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ECLECTIC  EDUCATIONAL  SERIES. 


ORAL    LESSONS 


IN 


NUMBER 


A    MANUAL     FOR    TEACHERS 


E.  E.  WHITE,  A.  M.,  LL.  D. 
1 1 

Author  of  Series  of  Arithmetics,  School  JRecords,  Etc. 


VAN  ANTWERP,  BRAGG  &  CO. 

CINCINNATI  FEW  YORK 


ECLECTIC    EDUCATIONAL    SERIES. 


WHITE'S  NEW  ARITHMETICS. 

TWO-BOOK  SERIES. 

White's  New  Elementary  Arithmetic, 
White's  New  Complete  Arithmetic, 

Uniting  Oral  and  Written  Processes,  and  Embodying  the  Inductive  Method. 
Key  to  White's  New  Complete  Arithmetic. 


WHITE'S  NEW  ARITHMETICS  afford  a  varied  and  thorough  drill  In  all  fun- 
damental processes. 

They  contain  an  unusually  large  number  of  Miscellaneous  Review  Prob- 
lems. 

The  series  abounds  in  practical  applications  of  numerical  processes  to 
mercantile  business,  the  mechanic  arts,  etc. 

Oral  and  Written  Exercises  are  united  in  a  practical  and  philosophical 
manner. 

Another  important  feature  is  a  true  and  practical  embodiment  of  the  in- 
ductive method  of  teaching. 

The  Definitions,  Rules,  and  Statements  of  Principles  are  models  of  accu- 
racy, clearness,  and  conciseness. 

Full  Descriptive  Circulars  and  Price-List  on  Application. 


COPYRIGHT, 

1884, 

VAN   ANTWERP,    BRAGG  4   CO. 


ECLECTIC  PRESS: 

VAN  ANTWERP,   BRAGG  A  CO. 


PREFACE. 


THE  great  importance  of  the  oral  instruction  in  number 
given  during  the  first  three  years  of  school,  demands  that  it 
be  wisely  and  carefully  arranged.  It  should  be  not  only  the 
best  possible  for  pupils  at  this  age,  but  it  should  be  the  best 
possible  preparation  for  the  instruction  which  is  to  follow. 

No  one  who  has  ever  attempted  to  map  out  such  a  series 
of  lessons  in  number,  will  claim  that  this  work  can  be  satis- 
factorily done  by  the  overburdened  teachers  in  our  primary 
schools.  The  attempt  to  do  this  has  been  made  on  a  wide 
scale,  and  the  results  have  not  been  satisfactory. 

It  is  not  difficult  for  primary  teachers  to  turn  the  crank  of 
such  a  number-teaching  machine  as  the  so-called  Grube 
method  may  be  made,  and  especially  when  abstract  numbers 
constitute  the  daily  grist.  It  is  also  easy  to  drill  pupils  for 
weeks  in  counting  to  100  by  ones,  by  twos,  etc.,  but  it  seems 
unnecessary  to  add  that  this  monotonous  drill  is  not  the  oral 
instruction  in  number  which  should  be  given  the  first  years 
of  school. 

It  is  believed  that  primary  teachers  will  welcome  the  guid- 
ance of  a  skillfully  arranged  series  of  oral  lessons  in  number — 
lessons  beginning  at  the  first  step  and  extending  through  the 
entire  oral  course.  The  careful  study  of  such  illustrative  les- 
sons will  enable  them,  as  a  class,  to  give  much  more  system- 
atic and  effective  oral  instruction  in  number  than  is  possible 

in  the  absence  of  such  needed  assistance, 
(iii) 


IV 


PREFACE. 


The  oral  lessons  in  number  presented  in  this  manual  are 
the  result  of  an  earnest  effort  to  supply  this  need.  They  are 
illustrative  lessons  for  the  guidance  of  teachers.  The  exercises  are 
given  in  detail,  to  indicate  clearly  the  nature  and  scope  of 
the  instruction  which  should  be  given  each  year. 

These  illustrative  lessons  are  based  on  the  fundamental 
principle  in  teaching,  that  all  primary  ideas  and  processes 
must  be  made  clear  and  familiar  before  any  successful  advance 
can  be  made.  They  also  recognize  the  important  fact  that 
children  acquire  primary  ideas  and  processes  very  slowly.  A 
failure  to  observe  this  principle  and  this  fact  is  one  of  the 
most  common  errors  in  teaching  the  elements  of  arithmetic. 

This  manual  also  contains  numerous  blackboard  and  slate 
exercises  to  accompany  the  oral  lessons;  a  concise  statement 
of  the  principles  involved  in  the  first  lessons  in  number ;  and 
suggestive  methods  of  teaching  elementary  processes,  defini- 
tions, and  rules.  It  is  designed  to  be  a  manual  of  elementary 
instruction  in  number. 

CINCINNATI,  0.,  Nov.  20,  1884. 


CONTENTS. 


PRINCIPLES  AND  EXPLANATIONS. 

PAGE 

Place  and  Extent  of  Oral  Instruction 

First  Year's  Instruction  in  Number 

Use  of  Abstract  Numbers * 

The  Teaching  of  the  Figures 9 

Counting  by  Ones 10 

Second  Year's  Instruction  in  Number 11 

The  Grube  Method 12 

Third  Year's  Instruction  in  Number 15 

Method  of  Teaching  Multiplication 16 

Method  of  Teaching  Division   .               17 


FIRST-YEAR  COURSE. 


Aims  and  Steps 19 

Appliances 20 

Lessons  for  Teaching  Numbers  1  to  10 21 

Review  Exercises  in  Addition,  Subtraction,  and  Analysis      ...  58 


SECOND-YEAR  COURSE. 

Aims  and  Steps 63 

The  Teaching  of  the  Numbers  11  to  15 64 

Lessons  in  Addition,  Subtraction,  and  Analysis 67 

The  Teaching  of  the  Numbers  16  to  20 88 

Lessons  in  Addition,  Subtraction,  and  Analysis 91 

Primary  Combinations  in  Addition Ill 

Analyses  of  the  Numbers  11  to  20 113 

(v) 


vi  CONTENTS. 

THIRD- YEAR  COURSE. 

ORAL  LESSONS  IN  MULTIPLICATION  AND  DIVISION. 

PAGE 

Aims  and  Steps 115 

The  Teaching  of  the  Numbers  20  to  100 116 

Lessons  in  Multiplication  and  Division 119 

Primary  Combinations  in  Multiplication 148 

Parts  of  Numbers 150 

Supplemental  Drills  in  Rapid  Combinations 153 

SUPPLEMENTAL  BLACKBOARD  EXERCISES. 

I.— Addition 155 

II.— Subtraction 158 

III.— Multiplication 160 

IV.— Division 161 

MISCELLANEOUS  LESSONS  AND  SUGGESTIONS. 

United  States  Money 163 

Common  Measures 166 

Measures  of  Liquids 166 

Dry  Measures 168 

Measures  of  Length  or  Distance 169 

Measures  of  Time 170 

Measures  of  Weight 172 

Metric  Measures 173 

Subtraction  :  Written  Process 175 

Long  and  Short  Division 178 

Place  of  the  Quotient 178 

The  Determining  of  Quotient  Figures 180 

Oral  Solutions 181 

Written  Processes 182 

Rules 183 

Definitions 186 

Summary  of  General  Process 187 


ORAL  LESSORS  IN  DUMBER 


PKINCIPLES  AND  EXPLANATIONS. 

THE  instruction  in  number  which  should  be  given 
during  the  first  two  years  of  a  school  course,  is  pri- 
marily objective,  and  can  best  be  given  orally.  When 
this  objective  oral  instruction  is  completed,  pupils  are 
prepared  to  use  an  elementary  arithmetic  with  advan- 
tage, and  hence  the  oral  instruction  of  the  third  year 
should  introduce  and  accompany  the  lessons  in  the 
book  used  by  the  pupils.  Nothing  is  gained  by  con- 
tinuing exclusive  oral  instruction  in  number  beyond 
the  second  school  year. 

The  putting  of  an  elementary  arithmetic  into  the 
hands  of  pupils  the  third  year,  not  only  increases 
their  interest,  and  otherwise  promotes  their  progress 
in  number,  but  it  greatly  relieves  the  teacher  of  un- 
necessary labor— not  an  unimportant  consideration. 

The  use  of  a  book  the  third  year,  and  subsequently, 
is  also  a  physical  advantage  to  pupils.  At  this  time  in 
the  school  course,  the  lessons  in  language  and  other 
written  exercises,  many  of  which  involve  the  copying 
of  exercises  from  the  blackboard,  are  a  severe  tax  on 
the  eyes  and  nerves  of  young  children.  It  is  feared 
that  children  are  sometimes  injured  by  the  large 
amount  of  copying  from  the  blackboard,  and  other 
slate  work,  required  of  them  during  the  first  three  or 
four  years  of  school. 

(vii) 


ORAL  LESSONS  IN  NUMBER, 


First  Year. 

The  aim  of  the  lessons  in  number  mapped  out  for 
the  first  year,  is  to  teach  objectively  the  numbers  from 
one  to  ten  inclusive — the  digital  or  primary  numbers. 

The  successive  exercises  include  numbering,  combin- 
ing, separating,  and  taking  away  groups  of  objects,  no 
group  exceeding  ten,  and  the  making  of  the  digital 
figures. 

The  exercises  in  numbering  are  intended  to  develop 
the  power  to  recognize  at  sight,  without  counting,  the 
number  of  objects  in  any  group  not  exceeding  ten — a 
power  essential  to  the  easy  mastery  of  the  other  exer- 
cises. It  is  claimed  by  primary  teachers  of  wide  expe- 
rience that  the  majority  of  children,  when  they  first 
enter  school,  can  not  give  at  sight  the  number  of  ob- 
jects in  a  group  exceeding  three.  A  few  weeks  of  drill 
will,  however,  enable  them  to  number  instantly  any 
group  not  exceeding  ten.  Dr.  Thomas  Hill,  ex-Presi- 
dent of  Harvard  College,  says  that  fairly  bright  chil- 
dren will  readily  learn  to  number  at  sight  as  many  as 
twelve  to  fifteen  objects — a  statement  based  on  actual 
experiment.  This  may  be  done  by  an  unconscious 
separation  of  the  larger  groups  into  two  smaller  groups, 
and  the  combining  of  these  ;  but,  howsoever  done,  the 
act  is  practically  instantaneous. 

The  perceptive  power  necessary  to  number  a  group 
of  objects  from  one  to  ten  at  sight,  is  not  only  funda- 
mental in  teaching  objectively  the  combining  and  sep- 
arating of  numbers,  but  it  is  also  of  great  value  in 
practical  life.  Nearly  every  person  has  daily  use  for 
the  power  to  recognize  at  sight  the  number  of  objects 
in  small  groups. 

The  purely  objective  exercises,  stated  above,  should 
be  followed  by  exercises  in  combining,  subtracting,  and 


PRINCIPLES  AND  EXPLANATIONS.  9 

separating  groups  of  objects  not  in  sight,  but  easily 
imagined.  This  step  should  be  taken  only  when  pupils 
have  acquired  the  skill  to  combine  and  separate  groups 
of  objects  in  sight.  The  true  order  is  first  the  number- 
ing, combining,  and  separating  of  groups  of  objects  in 
sight;  and,  second,  the  combining  and  separating  of 
groups  of  objects  not  in  sight. 

This  step  may  be  followed  by  the  adding,  subtract- 
ing, and  separating  of  the  corresponding  abstract  num- 
bers, but  so  strong  is  the  tendency  of  teachers  to  use 
abstract  numbers  to  the  neglect  of  needed  objective 
exercises,  that  it  is  believed  to  be  best  to  exclude  abstract 
numbers  entirely  from  the  first  year's  course.  There  is  no 
danger  that  the  use  of  abstract  numbers  will  be  omitted 
or  neglected  in  the  succeeding  years. 

This  tendency  of  teachers  to  use  abstract  numbers  in 
primary  lessons  in  arithmetic  is  largely  due  to  the  fact 
that  it  is  easier  to  drill  pupils  on  words  and  other  sym- 
bols than  it  is  to  teach  them  real  knowledge — a  fact  sadly 
illustrated  in  the  memoriter,  word,  and  figure  drills 
which  have  so  long  characterized  school  instruction. 

It  may  be  a  question  whether  children  should  be 
taught  the  figures  and  the  making  of  them  in  connec- 
tion with  these  first  year  lessons.  This  doubtless  de- 
pends on  the  age  of  the  children  taught.  If  children 
are  admitted  to  school  as  early  as  five  years  of  age,  the 
teaching  of  figures  may  be  wisely  deferred  until  the 
second  year,  or,  at  least,  until  the  latter  part  of  the  first 
year.  When  pupils  enter  school  at  the  age  of  six  years 
and  upwards,  the  figures  may  be  taught  at  the  close  of 
the  different  series  of  lessons,  but  even  then  it  may  be 
well  to  defer  such  instruction  for  a  few  v- 

The  skill  acquired  in  making  figures  the  first  year 
will  promote  the  progress  of  the  pupils  the  second  year, 
and  there  seems  to  be  little  danger  that  the  teaching  of 
the  figures  in  this  manner  will  lead  pupils  into  the 


10  ORAL  LESSONS  IN  NUMBER. 

error  of  confounding  figures  with  the  numbers  which 
they  represent— an  error  common  among  pupils  who, 
from  the  first,  use  figures  as  actual  numbers. 

The  first  step  in  teaching  a  number  is  to  develop  an 
idea  of  the  number  itself,  and  this  can  only  be  done  by 
objects.  A  number  is  neither  a  word  nor  a  figure,  and 
hence  it  can  not  be  taught  by  teaching  its  name  or  the 
figure  or  figures  that  express  it.  A  child  may  learn  the 
names  of  the  numbers  not  only  from  one  to  ten,  but 
from  one  to  one  hundred,  and  not  have,  as  a  result,  a 
clear  idea  of  a  single  number  named.  It  is  a  great 
error  to  infer  from  the  fact  that  the  idea  of  number 
may  be  intuitive,  that  an  idea  of  the  different  primary 
numbers  need  not  be  developed  and  taught.  The  idea 
of  space  is  intuitive,  but  it  does  not  follow  that  such 
particular  space  concepts  as  the  line,  the  triangle,  the 
square,  the  circle,  the  ellipse,  the  cube,  the  sphere,  etc., 
are  innate.  Experience  shows  that  a  clear  idea  of  these 
primary  space  or  form  concepts  is  slowly  acquired  by 
children,  and  that  they  need  to  be  carefully  taught. 
For  a  like  reason,  the  primary  numbers  should  be  care- 
fully taught,  and  this  can  best  be  done  by  numbering, 
combining,  and  separating  groups  of  objects. 

Special  attention  is  called,  in  this  connection,  to  the 
importance  of  avoiding  in  these  primary  processes  the 
too  common  practice  of  counting  by  ones.  The  number- 
ing, combining,  and  separating  of  groups  of  objects  by 
counting,  leads  to  the  pernicious  habit  of  adding  and 
subtracting  numbers  by  counting — a  habit  that  must  be 
overcome  before  a  pupil  can  learn  to  add  or  subtract 
numbers  as  wholes.  When  a  child  can  number  a  group 
of  three  objects  at  sight,  he  should  be  taught  a  group 
of  four  objects,  as  three  and  one,  or  one  more  than 
three,  and  not  simply  as  four  ones.  It  is  not  only  un- 
necessary to  number  four  objects  by  counting  one,  two, 
three,  four,  but  this  counting  is  likely  to  give  the  child 


PRINCIPLES  AND  EXPLANATIONS.  11 

the  erroneous  idea  that  the  first  object  is  one,  the  second 
two,  the  third  three,  the  fourth  four.  The  child  must 
see  the  entire  group  as  four  objects,  and  when  he  has 
learned  that  four  objects  are  three  objects  and  one 
object,  or  two  objects  and  two  objects,  he  has  a  clear 
idea  of  the  number  four. 

The  same  is  true  of  combining  and  separating  groups 
of  objects.  The  child  has  not  learned  to  add  3  balls 
and  4  balls,  for  example,  until  he  sees  7  balls  the 
instant  3  balls  and  4  balls  are  presented  to  the  mind. 
The  easy  and  quick  perception  of  the  sum  of  any  two 
groups  of  objects,  present  or  imagined,  each  not  exceed- 
ing ten,  is  the  first  step  in  the  art  of  adding  and  sub- 
tracting numbers. 

It  is  admitted  that  first-year  pupils  can  be  taken  over 
more  ground  than  is  covered  by  these  primary  lessons, 
but  it  is  believed  that  nothing,  in  the  end,  will  be 
gained  by  adding  instruction  that  can  be  better  given 
the  second  and  third  years.  A  good  maxim  in  primary 
teaching  is,  u  Make  haste  slowly" 


Second  Year. 

The  several  series  of  exercises  that  constitute  the 
second  year's  course  of  oral  instruction  in  number, 
involve  and  use  the  ideas  and  skill  acquired  by  pupils 
the  first  year,  and  hence  there  is  less  occasion  for  the 
use  of  visible  objects.  In  teaching  the  numbers  from 
eleven  to  twenty  inclusive,  present  and  imagined  ob- 
jects are  successively  used ;  and  the  processes  of  add- 
ing, subtracting,  and  separating  numbers  are  each 
introduced  by  objects  in  sight  or  easily  imagined.  The 
order  observed  is  as, follows: 

1.  The  combining,  taking  away,  and  separating  of 
groups  of  objects  in  sight  or  easily  imagined. 


12  ORAL  LESSONS  IN  NUMBER. 

2.  The   adding,   subtracting,   and    analyzing  of  con- 
crete numbers. 

3.  The  adding,  subtracting,  and  analyzing  of  abstract 
numbers. 

The  twofold  aim  of  all  the  exercises  in  the  first  and 
second  years  is  to  impart  a  clear  idea  of  the  numbers 
from  one  to  twenty,  and  to  give  the  pupil  the  power 
to  add  and  subtract  the  primary  or  digital  numbers 
without  counting.  The  entire  series  of  exercises  recog- 
nizes the  fact  that  the  power  to  perceive  the  sum  or 
difference  of  any  two  digital  numbers  is  the  basis  of 
the  art  of  rapid  and  accurate  computation.  If  this 
power  be  acquired  the  first  two  years  of  school,  the 
time  devoted  to  the  teaching  of  number  will  be  wisely 
and  profitably  employed. 

It  is  at  this  point  that  the  system  of  instruction  in 
number,  presented  in  this  manual,  differs  most  widely 
from  the  so-called  Grube  method,  which,  from  the  first, 
unites  the  four  processes  of  addition,  subtraction,  mul- 
tiplication, and  division,  as  is  shown  in  the  following 
scheme  for  teaching  the  number  seven  to  first-year 
pupils : 

6+1,  1  +  6,  7—1,  7  — 6;5  +  2,  2  +  5,  7  —  2,  7-5; 
4  +  3,  3  +  4,  7-3,  7  —  4;  II  7x1,  7-5-1,  }  of  7;  2  X 
3  +  1,  7^2,  i  of  7;  3x2  +  1,  7-5-3,  I  of  7;  4x  1  +  3, 
7-f-4,  i  of  7;  5X1  +  2,.  7-5,  I  of  7;  6x1  +  1,  7  -4- 
6,  J-  of  7. 

It  is  seen  that  the  exercises  which  precede  the  vertical 
lines  (II)  involve  addition  and  subtraction,  and  that 
those  which  follow  these  lines  involve  multiplication 
and  division.  Is  there  any  such  immediate  and  neces- 
sary connection  between  the  ^concepts  and  processes  of 
addition  and  subtraction  and  those  of  multiplication 
and  division  as  requires  the  teaching  of  these  four  pro- 
cesses together?  The  concepts  and  processes  of  addi- 


PRINCIPLES  AND  EXPLANATIONS.  13 

tion  arid  subtraction  relate  to  numbers  as  composed  of 
parts,  and,  being  inverse  processes,  should  be  taught  to- 
gether. The  concepts  and  processes  of  multiplication 
and  division  relate  to  numbers  as  composed  of  factors, 
and,  being  inverse  processes,  should  likewise  be  taught 
together.  But  there  is  nothing  in  the  relation  of  these 
two  sets  of  inverse  processes  to  each  other  that  necessi- 
tates or  justifies  the  teaching  of  them  from  the  first  as 
correlates.  On  the  contrary,  there  are  strong  reasons 
against  the  mixing  up  of  these  two  sets  of  relations  in 
the  child's  first  lessons  in  number. 

When  the  concepts  and  processes  of  addition  and 
subtraction  are  familiar  to  pupils,  those  of  multiplica- 
tion and  division  are  easily  acquired.  A  knowledge  of 
the  former  assists  in  acquiring  the  latter.  Addition, 
for  example,  assists  the  pupil  in  determining  the  pro- 
duct of  two  digits,  and  the  more  familiar  the  pupil  is 
with  the  process  of  addition,  the  more  easily  will  he 
learn  multiplication.  On  the  contrary,  multiplication 
can  render  a  child  little,  if  any,  assistance  in  learning 
the  sum  of  two  digits.  In  the  order  of  acquisition,  the 
processes  of  multiplication  and  division  follow  those  of 
addition  and  subtraction,  and  there  is  nothing  gained 
by  alternating  these  two  sets  of  inverse  processes  in  the 
first  lessons  in  number. 

It  is  admitted  that  these  four  processes  can  be  taught 
simultaneously  to  children  five  years  of  age,  and  even 
without  using  objects.  Primary  teachers  have  accom- 
plished even  more  difficult  things,  as  the  history  of 
primary  instruction  sadly  attests.  Young  children  have 
been  taught  to  spell  orally  many  hundreds  of  words, 
most  of  which  expressed  no  idea  whatever  to  the 
speller.  Many  a  child  has  committed  the  multiplica- 
tion table  before  he  could  add  7  and  7,  and  hosts  of 
children  have  learned  to  repeat  pages  of  the  text  in 
their  books  without  clearly  comprehending  a  sentence 


14  ORAL  LESSONS  IN  NUMBER 

repeated.  The  question  is  not  whether  young  chil- 
dren can  do  these  things.  The  more  vital  question 
is,  "  Is  this  training  the  best  possible  for  young  chil- 
dren?" 

A  child  might  possibly  be  taught  to  walk  by  being 
put  through  daily  a  drill  which  would  call  into  play, 
in  succession,  all  the  muscles  in  his  legs,  and  give  to 
each  every  possible  variety  of  movement.  Such  a 
method  might  even  claim  to  be  "  scientific,"  but 
nature's  method  of  teaching  a  child  to  walk  is  to 
induce  it  to  take  one  step,  then  two,  and  so  on,  in 
walking,  and  the  process  can  not  be  hurried,  strength 
and  skill  in  walking  being  acquired  slowly. 

The  primary  and  fundamental  processes  in  number 
are  ADDITION  and  SUBTRACTION,  and  the  natural  way  to 
teacli  a  child  to  add  and  subtract  numbers  is  to  give 
him  exercises  involving  these  processes.  Exercises  in  mul- 
tiplying and  dividing  numbers  can  render  no  assistance 
in  these  first  lessons;  and,  if  they  could,  such  assistance 
is  not  needed,  since  the  processes  of  addition  and  sub- 
traction are  easily  taught  without  it. 

It  may  be  true  that  a  child's  knowledge  of  a  given 
number  is  not  perfect  until  he  has  viewed  it  in  all  possi- 
ble relations  to  other  numbers.  A  child's  "  grasp  "  of  the 
size  of  numbers,  exceeding  say  fifteen,  may  not  be  per- 
fect until  he  has  compared  them  with  the  primary  or 
digital  numbers,  both  with  reference  to  their  difference 
and  to  their  quotient,  but  it  does  not  follow  that  both 
of  these  comparisons  should  be  made  in  the  first  lessons 
in  number.  It  may  be  wisely  taken  for  granted  that 
the  third  and  subsequent  years  of  arithmetical  instruc- 
tion will  do  something  to  widen  the  pupil's  grasp  of 
numbers.  It  is  seriously  questioned  whether  a  little 
child's  grasp  of  the  number  7  would  be  much  broad- 
ened by  the  series  of  exercises  which  follow  the  vertical 
"lines  in  the  scheme  above  presented. 


PRINCIPLES  AND  EXPLANATIONS.  15 


Third  Year. 

The  preceding  exercises  in  adding  groups  of  objects 
and  numbers  and  in  separating  groups  of  objects  and 
numbers  into  parts,  have  prepared  the  way  for  the  easy 
mastery  of  the  primary  processes  of  multiplication  and 
division. 

It  seems  important  here  to  note  that  the  adding  of 
equal  numbers  is  not  multiplication,  and  that  the  sepa- 
rating of  a  number  into  equal  parts  is  not  arithmetical 
division.  The  fact  that  the  processes  of  addition  and 
multiplication  are  not  the  same,  may  be  clearly  shown 
by  adding  say  twenty-four  25's  and  by  multiplying  25 
by  24.  While  the  results  reached  are  numerially  the 
same,  the  two  processes  are  obviously  not  the  same. 
The  distinction  between  dividing  a  number  into  equal 
parts  and  arithmetical  division  may  be  shown  by  sepa- 
rating, say,  15  blocks  into  5  equal  parts,  and  by  dividing 
15  cents  by  3  cents.  In  the  first  process,  there  is  neither 
a  numerical  divisor  nor  a  numerical  quotient ;  in  the 
second  process,  5  cents  is  the  divisor,  or  measure,  and  3 
is  the  quotient.  The  first  process  divides  15  blocks 
into  parts,  which  is  division  only  in  the  primary  mean- 
ing of  the  term ;  the  second  process  divides  15  cents  by 
&  factor  (5  cents)  and  finds  the  other  factor  (3),  and  this 
is  arithmetical  division. 

It  is  thus  seen  that  the  processes  of  adding  and  sepa- 
rating numbers,  previously  taught,  deal  with  numbers 
as  composed  of  parts,  but  the  processes  of  multiplica- 
tion and  division  now  to  be  taught  deal  with  numbers 
as  composed  of  factors.  In  multiplication  two  numbers 
are  given,  and  it  is  required  to  find  their  product;  in 
division  two  numbers  are  given,  and  it  is  required  to 
find  how  many  times  one  of  these  numbers  is  con- 
tained in  the  other; — that  is,  a  product  and  one  of  its 
factors  are  given,  and  it  is  required  to  find  the  other 


1G  ORAL  LESSONS  IN  NUMBER. 

factor.  These  two  processes  are  fundamental  in  arith- 
metical computations,  and  their  thorough  mastery  is  of 
prime  importance. 

The  final  end  to  be  attained  in  the  teaching  of  multi- 
plication, is  to  impart  to  the  pupil  the  ability  to  per- 
ceive the  product  of  any  two  digital  numbers,  without 
adding,  as  soon  as  they  are  presented  to  the  mind. 
The  perception  of  these  products  must  be  as  direct  and 
as  instantaneous  as  the  recognition  of  the  most  familiar 
words  in  reading.  When  this  ability  is  acquired,  the 
pupil  is  prepared  to  multiply  numbers  expressed  by 
two  or  more  figures. 

How  should  the  multiplication  of  the  primary  or  digital 
numbers  be  taught? 

The  old  method,  which  required  pupils  to  commit  to 
memory  a  table  of  all  the  products  of  the  digital  num- 
bers, two  by  two,  is  not  satisfactory.  Many  children 
have  great  difficulty  in  memorizing  and  retaining  so 
many  products ;  and,  when  memory  fails  them,  their 
only  recourse  is  a  reference  to  the  printed  multiplica- 
tion table — a  great  hindrance  and  a  great  inconvenience. 
It  is  believed  that  full  three  fourths  of  the  pupils  who 
mechanically  memorize  the  multiplication  table,  are 
never  able  to  multiply  large  numbers  without  more  or 
less  dependence  on  the  printed  table. 

A  better  method  is  to  teach  pupils  how  to  determine 
the  product  of  any  two  digits  by  adding  before  requiring 
them  to  commit  the  product  to  memory.  When  pupils 
see  that  the  product  of  4  and  5,  for  example,  is  the 
same  numerically  as  the  sum  of  four  5's  or  five  4's, 
they  not  only  have  a  clear  idea  of  the  value  of  this 
product,  but  they  have  a  key  to  it,  on  which  they  can 
depend  if  memory  should  ever  fail  to  recall  it.  Be- 
sides, this  method  of  learning  the  products  assists  in 
fixing  them  in  memory,  and  the  labor  involved  in 
memorizing  the  multiplication  table  is  greatly  reduced. 


PRINCIPLES  AND  EXPLANATIONS.  17 

It  should,  however,  be  kept  in  mind  that  the  finding 
of  the  numbers  corresponding  to  these  products  by  ad- 
dition is  an  introductory  step,  and  should  be  discon- 
tinued as  soon  as  its  object  is  attained.  The  next  and 
essential  step  is  to  teach  the  products  of  the  digital 
numbers  as  products,  and  as  such  to  fix  them  in  the 
memory.  Memory  drills  should  be  continued  until 
any  two  of  the  digital  numbers  are  as  clearly  and  as 
directly  associated  with  their  product  as  with  their 
sum. 

Division  should  be  taught  as  the  inverse  of  multiplication. 

The  fact  that  3  times  4  or  4  times  3  is  12,  involves 
the  facts  that  4  is  in  12  three  times,  and  3  in  12  four 
times;  and  when  these  facts  are  taught  in  connection 
with  each  other,  the  pupil  sees  the  latter  in  the  former, 
and  is  thus  relieved  of  the  necessity  of  committing  the 
division  results  to  memory. 

There  is  nothing  gained  by  the  attempt  to  teach 
division  as  a  method  of  subtraction.  It  is  not  a 
method  of  subtraction.  It  is  true  that  the  quotient 
shows  how  many  times  the  divisor  may  be  subtracted 
from  the  dividend,  but  this  quotient  is  not  found  by 
subtraction.  This  may  be  clearly  shown  by  dividing 
125  by  25,  and  by  subtracting  25  from  125  as  many 
times  as  possible.  The  dissimilarity  of  the  two  pro- 
cesses is  too  obvious  to  require  a  formal  comparison  of 
them,  and  it  is  also  as  obvious  that  the  results  are  not 
the  same.  The  final  result  of  the  several  subtractions 
of  25  from  125  is  0,  and  it  is  only  by  counting  (or  in- 
spection) that  the  number  of  subtractions  is  deter- 
mined. The  quotient  5,  obtained  by  division,  is  the 
number  of  times  25  is  contained  in  125,  and,  as  a  con- 
sequence, it  shows  that  25  'can  be  taken  from  125  five 
times.  This  consequence  is  an  interesting  fact,  but  it 
does  not  make  division  a  method  of  subtraction.  The 
fact  that  there  are  but  two  fundamental  principles  in 

O.  L.-2. 


18  ORAL  LESSONS  IN  NUMBER. 

number,  synthesis,  and  analysis  does  not  show  that 
there  are  but  two  fundaniental  processes.  In  practice 
there  are  two  synthetic  processes, — addition  and  mul- 
tiplication ;  and  two  analytic  processes, —  subtraction 
and  division.  Addition  and  subtraction  synthesize 
and  analyze  numbers  as  parts,  and  are  inverse  pro- 
cesses. Multiplication  and  division  synthesize  and 
analyze  numbers  as  factors,  and  are  likewise  inverse 
processes. 


FIRST-YEAR   COURSE. 


Aim. 

To   teach  objectively  the   numbers  from  one   to    ten  in- 
clusive. 

Steps. 

1.  The  numbering  of  the  objects  in  any  group  not 
exceeding  ten,  without  counting. 

2.  The  combining  of  any  two  groups  whose  sum  does 
not  exceed  ten,  without  counting. 

3.  The  taking  from  any  group  not  exceeding  ten  each 
of  the  two  smaller  groups  combined  to  form  it. 

4.  The  separating  of  any   group   not  exceeding  ten 
into  the  two  smaller  groups  that  compose  it,  and  then 
taking  successively  each  of  the  two  smaller  groups,  thus 
found,  from  the  original  group. 

5.  The    combining,    separating,    and   subtracting   of 
groups  of  objects  not  in  sight  but  easily  imagined, — 
groups  not  exceeding  ten. 

6.  The  comparing  of  two  groups  of  objects,  in  sight 
or  imagined,  to  see  how  much  one  group  is  greater  or 
less  than  the  other. 

7.  The  applying  of  the  processes  learned  to  the  solu- 
tion of  easy  problems  involving  a  simple  exercise  of 
the  imagination  and  the  judgment. 

8.  The  teaching  of  the  figures  that  express  the  digital 
numbers. 

(19) 


20  ORAL  LESSONS  IN  NUMBER. 

NOTE. — In  teaching  the  successive  numbers,  it  is  not  im- 
portant that  these  steps  be  taken  uniformly  in  the  order  above 
given.  The  second  and  third  steps  may  properly  be  combined 
in  one  step  as  inverse  processes.  The  first  process  in  the  fourth 
step  may  first  be  given  separately,  and  then  the  two  processes 
may  be  combined  in  one,  as  given. 

Appliances. 

In  order  to  teach  objectively  the  following  lessons, 
the  teacher  should  be  supplied  with  a  variety  of  ob- 
jects of  convenient  sizes  and  of  different  shapes  and 
colors,  such  as  counters,  small  rods,  blocks,  shells,  peb- 
bles, buttons,  beans,  etc. 

The  objects  used  by  the  teacher  in  class  instruction 
should  be  large  enough  to  be  easily  seen  by  all  the  pu- 
pils, as  blocks,  shells,  large  buttons,  pebbles,  and  rods 
of  about  the  size  of  pencils  or  common  pen-holders. 
The  pupils  can  use  at  their  seats  such  small  objects  as 
match-sticks,  tooth-picks,  small  buttons,  flat  beans,  etc. 

The  numeral  frame  can  also  be  used  by  the  teacher, 
but  the  balls  are  usually  too  small  for  use  before  large 
classes.  A  better  aid  would  be  two  wires,  with  ten 
balls  on  each,  the  balls  being  from  one  to  two  inches  in 
diameter,  and  the  wires  being  stretched  parallel  and 
near  each  other,  and  in  such  a  position  that  the  teacher 
can  readily  move  the  balls  with  a  pointer  or  with  the 
hand.  One  wire  with  ten  balls  can  be  used  the  first 
year,  but  there  are  objections  to  the  exclusive  use  of 
objects  of  one  kind,  and  especially  to  the  presenting  of 
objects  uniformly  in  lines  or  rows.  The  exclusive  use 
of  ten  balls  on  a  single  wire  would  give  many  young 
pupils  the  idea  that  numbered  objects  stand  in  rows, 
and  combine  and  separate  in  straight  lines.  The  groups 
of  objects  numbered,  combined,  and  separated  by  first- 
year  pupils  should  be  presented  in  all  sorts  of  shapes — 
with  the  irregularity  of  groups  in  nature. 


FIRST-YEAR   COURSE.  21 

Much  interest  may  be  imparted  to  these  objective 
drills  by  using,  in  their  season,  flowers  and  fruits  of 
different  kinds.  Beautiful  roses,  apple-blossoms,  pan- 
sies,  dandelions,  etc.,  are  easily  gathered  or  secured ;  also, 
apples,  plums,  peaches,  cherries,  apricots,  etc.  A  little 
pains  will  secure  the  needed  variety  of  objects  for  class 
instruction. 

The  objects  used  in  class  instruction  (balls  and  other 
round  objects  excepted),  should  rest  on  a  surface  in- 
clined enough  to  show  all  the  objects  well,  and,  at  the 
same  time,  so  nearly  level  that  the  objects  used  will 
not  slide  or  roll.  For  this  purpose,  a  large  slate  or  a 
light  board  of  convenient  size  may  be  placed  on  a  table, 
or  the  table  itself  may  be  inclined  a  little  by  slipping 
blocks  under  two  of  the  legs. 


LESSON  I. 

TJie  Numbers  1  and  2. 

NOTE. — Nearly  all  children  know  the  numbers  one  and  two 
when  they  enter  school,  and  hence  but  little  time  need  be  spent 
in  teaching  these  numbers. 

1.  How  many  hands  do  I  hold  up?  (Raise  right 
hand.)  How  many  hands  do  I  now  hold  up?  (Raise 
both  hands.) 

Hold  up  your  right  hand.  Hold  up  your  left  hand. 
Hold  up  both  hands.  How  many  hands  do  you  now 
hold  up?  "Two  hands." 

How  many  books  do  I  hold  up?  "One  book."  How 
many  now?  "Two  books."  Hold  up  one  thumb. 
Hold  up  two  thumbs. 

How  many  eyes  have  you?  How  many  ears?  How 
many  chins  ?  How  many  cheeks  ?  How  many  tongues  ? 
How  many  feet? 


22  ORAL  LESSONS  IN  NUMBER. 

2.  Mary  may  bring  me  one  book,  and  Kate  may 
bring  me  one.  How  many  books  have  I  now? 

How  many  books  are  one  book  and  one  book? 

George  may  put  one  block  on  the  table,  and  Charles 
may  put  one  block  with  George's.  How  many  blocks 
are  on  the  table? 

Frank  may  take  away  one  of  the  blocks.  How  many 
blocks  are  left  on  the  table? 

How  many  blocks  are  one  block  and  one  block? 
Two  blocks  less  one  block? 

How  many  boys  are  one  boy  and  one  boy?  Two 
boys  less  one  boy? 


3.  Here  are  two  books,  and  I  now  separate  them  into 
one  book  and  one  book.     (Suit  action  to  word.) 

Two  blocks  are  one  block  and  block. 

Two  blocks  less  one  block  are  block. 


NOTE. — The  teaching  of  the  figures  in  this  and  succeeding 
lessons  may  be  postponed  several  weeks,  and  then  be  taught  in 
connection  with  a  review  of  these  lessons.  See  page  9. 

4,  This  is  the  word  that  stands  for  one:   @ne. 
This  is  the  figure  that  stands  for  one:  ]_. 

This  is  the  word  that  stands  for  two:  <<^>0. 
This  is  the  figure  that  stands  for  two :  J?. 
Make  the  figure  1  011  your  slate. 
Make  the  figure  1  on  -your  slate  twice. 
Make  the  figure  2  on  your  slate. 
Make  the  figure  2  on  your  slate  twice. 

NOTE. — Devote  an  entire  lesson  to  the  teaching  of  these  figures, 
and  give  the  pupils  practice  in  making  them  both  on  the  board 
and  on  the  slate.  If  the  slates  are  not  properly  ruled,  see  that 
this  is  done,  and  have  all  figures  made  neatly. 


FIRST- YEAR  COURSE.  23 

LESSON  II. 

The  Number  3. 

1.  How  many  hands  do  I  hold  up? 
"  Two  hands."     How  many  fingers  ?     "  Two 
fingers."     How  many  fingers  do  I  now  hold 
up?     "Three  fingers." 

Hold  up  two  fingers  on  your  left  hand.  Hold  up 
three  fingers  on  your  right  hand. 

NOTE. — The  fingers  are  convenient  objects  for  use  in  teaching 
the  digital  numbers,  but  they  should  not  be  used  exclusively. 
See  Lesson  IV,  note. 

Mary  may  put  two  blocks  on  the  table,  and  Kate 
may  put  one  block  with  Mary's.  How  many  blocks 
are  now  on  the  table? 

Jane  may  take  away  one  block.  How  many  blocks 
are  now  on  the  table? 

How  many  fingers  do  I  hold  up?  "Three  fingers." 
How  many  now  ?  "  Two  fingers."  How  many  books  ? 
"  Two  books."  How  many  books  now  ?  "  Three 
books." 


2.  How  many  balls  are  two  balls  and  one  ball?  One 
ball  and  two  balls  are  how  many  balls? 

How  many  acorns  are  two  acorns  and  one  acorn  ? 

How  many  books  are  one  book  and  one  book?  Two 
books  and  one  book?  One  book  and  two  books? 

John  may  put  two  buttons  and  one  button  on  the 
table.  How  many  buttons  are  on  the  table? 

Charles  may  take  away  two  of  the  buttons.  How 
many  buttons  are  left  on  the  table? 

How  many  buttons  are  three  buttons  less  two  but- 
tons? 


24  ORAL  LESSONS  IN  NUMBER. 

How  many  blocks  are  one  block  and  two  blocks? 
Three  blocks  less  one  block? 

How  many  pencils  are  three  pencils  less  two  pencils? 
Three  pencils  less  one  pencil? 

How  many  chairs  are  three  chairs  less  two  chairs? 
Three  chairs  less  one  chair?  Two  chairs  less  one  chair? 

ffO.^       ,±t-   -..- 


3.  Three  books  are  two  books  and  book,  or  one 

book  and  books. 

NOTE. — Separate  the  three  books  into  two  groups,  and  let  the 
pupils  see  that  three  books  are  two  books  and  one  book,  or  one 
book  and  two  books. 

John  may  put  three  blocks  on  the  table  and  divide 
them  into  two  groups.  How  many  blocks  in  each 
group  ? 

Three  blocks  are  two  blocks  and  block,  or  one 

block  and  blocks. 

Three  blocks  less  two  blocks  are  block;  three 

blocks  less  one  block  are  blocks. 

Three  books  are  one  book,  one  book,  and book. 

Kate  may  put  three  blocks  on  the  table  and  divide 
them  into  three  parts.  How  many  blocks  in  each  part  ? 
"One  block." 


4.  How  many  pigs  are  two  pigs  and  one  pig?  Three 
pigs  less  one  pig? 

How  many  cents  are  one  cent  and  two  cents  ?  Three 
cents  less  one  cent?  Two  cents  less  one  cent? 

John  may  hold  up  three  fingers,  and  James  two 
ringers.  How  many  more  ringers  does  John  hold  up 
than  James? 


FIRST-YEAR  COURSE.  25 

Jane  may  put  three  buttons  and  two  pebbles  on  the 
table.  How  many  more  buttons  than  pebbles  on  the 
table  ? 

Susan  bought  three  sticks  of  candy  and  ate  one  stick. 
How  many  sticks  of  candy  had  she  left? 

Charles  picked  three  pears  and  gave  two  of  them  to 
his  mother.  H-^  -  pears  had  he  left? 


5.   This   is   the   word    that    stands    for   the   number 

three :   <^^% 

This  is  the  figure  that  stands  for  three:  <?. 
Make  the  figure  3  on  your  slate  three  times. 
Make  the  figures  1,  2,  3  on  your  slate. 


LESSON  III. 

The  Number  4- 

NOTE. — The  teacher  will  need  to  multiply  the  exercises  in  this 
and  the  following  lessons.  No  step  should  be  left  until  the  pupils 
can  take  it  easily.  It  will  often  be  necessary  to  devote  several 
class  exercises  to  one  of  the  subdivisions  of  a  lesson.  The  teacher 
should  also  vary  the  language  used,  making  it  less  formal  and 
more  conversational  and  lively. 

1.  How  many  fingers  do  I  hold  up  on 
my  right  hand?  How  many  on  my 
left  hand?  How  many  fingers  on  both 
hands  ? 

Hold  up  three  fingers.     Hold  up  four  fingers. 

How  many  books  do  I  hold  up?  "Three  books." 
How  many  now  ?  "  Four  books." 

How  many  pencils  in  my  right  hand?  "Four  pen- 
cils." How  many  in  my  left  hand?  "Two  pencils." 


26  ORAL   LESSONS  IN  NUMBER. 

Mary  may  put  three  blocks  on  the  table,  and  Kate 
may  put  one  block  with  Mary's.  How  many  blocks  are 
on  the  table? 

John  may  put  two  shells  on  the  table,  and  Charles 
may  put  two  shells  with  John's.  How  many  shells  are 
on  the  table? 

How  many  pencils  in  my  right  hand  ?  "  Four  pen- 
cils." How  many  pencils  in  my  left  hand?  "Three 
pencils." 

How  many  balls  do  I  move  on  the  upper  wire? 
"Three  balls."  How  many  on  the  lower  wire?  "Four 
balls." 


2.  How  many  blocks  are  three  blocks  and  one  block? 
One  block  and  three  blocks?  How  many  blocks  are 
two  blocks  and  two  blocks? 

Mary  may  put  three  shells  on  the  table,  and  Jane 
may  put  one  shell  with  Mary's.  How  many  shells  are 
on  the  table? 

How  many  shells  are  three  shells  and  one  shell? 

Charles  may  hold  up  two  fingers  on  his  right  hand 
and  two  fingers  on  his  left  hand.  How  many  fingers 
does  Charles  hold  up? 

How  many  fingers  are  two  fingers  and   two  fingers? 


3,  Jane  may  put  two  blocks  and  two  blocks  on  the 
table.  How  many  blocks  on  the  table? 

Kate  may  take  away  two  of  the  blocks.  How  many 
blocks  are  left  on  the  table? 

How  many  blocks  are  four  blocks  less  two  blocks? 

Susan  may  take  three  pencils  in  her  left  hand  and 
one  in  her  right  hand.  How  many  pencils  has  Susan  f 


FIRST-TEAR  COURSE.  27 

Susan  may  give  one  pencil  to  Mary.  How  many 
pencils  has  Susan  left? 

How  many  pencils  are  four  pencils  less  one  pencil? 
Four  pencils  less  three  pencils? 

How  many  birds  are  three  birds  and  one  bird?  Four 
birds  less  one  bird?  One  bird  and  three  birds?  Four 
birds  less  three  birds?  Two  birds  and  two  birds? 
Four  birds  less  two  birds? 

Willie  may  hold  up  four  fingers,  and  George  two 
fingers.  How  many  more  fingers  does  George  hold  up 
than  Willie? 


4.  Let  us  divide  four  shells  into  two  groups.  How 
many  shells  in  this  group?  "Three  shells."  How 
many  shells  in  this  group?  U0ne  shell." 

Four  shells  are  three  shells  and  -  —  shell,  or  one 
shell  and  shells. 

Let  us  now  divide  four  shells  into  two  equal  groups. 
How  many  shells  in  each  group?  "Two  shells." 

Four  shells  are  two  shells  and  shells. 

Kate  may  put  four  buttons  on  the  table  and  divide 
them  into  two  groups  in  two  ways. 

NOTE. — The  two  divisions  will  give  these  results: 

,-,       ,    .  /  3  buttons  and  1  button,  or  1  button  and  3  buttons. 

Four  buttons  are  j 2  buttons  and  2  buttons. 

John  may  divide  four  pebbles  into  two  groups  in  two 
ways. 

Four  birds  are  three  birds  and bird,  or  one  bird 

and  birds. 

Four  birds  less  one  bird  are  birds;  four  birds 

less  three  birds  are  bird. 

Four  birds  are  two  birds  and  -  -  birds;  four  birds 
less  two  birds  are  birds. 


28  ORAL  LESSONS  IN  NUMBER. 

5.  John  has  four  marbles  in  his  right  hand  and  three 
marbles  in  his  left  hand.  How  many  marbles  in  his 
right  hand  more  than  in  his  left? 

Kate  found  four  eggs  in  one  nest  and  two  eggs  in 
another.  How  many  more  eggs  did  she  find  in  the  first 
nest  than  in  the  second? 

Charles  picked  four  peaches  and  gave  two  of  them  to 
his  sister.  How  many  peaches  had  Charles  left? 

Mary  picked  three  roses  from  a  bush  and  one  rose 
from  another  bush.  How  many  roses  did  she  pick? 

How  many  cents  are  three  cents  and  one  cent?  Two 
cents  and  two  cents? 

How  many  cents  are  four  cents  less  three  cents? 
Four  cents  less  one  cent?  Four  cents  less  two  cents? 


6.  This  k  the  word  that  stands  for  four: 
This  is  the  figure  that  stands  for  four :  ^. 
Make  the  figure  4  011  your  slate  four  times. 
Make  the  figures  1,  2,  3,  4  on  your  slate. 

NOTE. — See  page  62,  note. 


LESSON  IV. 

The  Number  5. 

1.  How  many  fingers  do  I  hold  up  on 
my  right  hand?  How  many  on  my  left 
hand?  How  many  on  both  hands? 

NOTE. — In  giving  these  and  similar  finger  exercises,  put  down 
the  hands,  and,  as  the  question  is  asked,  raise  them  (one  or  both, 
as  the  case  may  be)  with  the  requisite  number  of  fingers  open, 
thus  presenting  the  group  of  fingers  to  the  eye  as  a  whole.  The 
number  of  fingers  presented  should  be  given  instantly  by  the 
pupils. 


FIRST-YEAR  COURSE.  29 

Hold  up  four  fingers.  How  many  fingers  more  will 
make  five  fingers  ?  Hold  up  five  fingers.  -  i 

George  may  put  four  blocks  on  the  table.  How 
many  blocks  must  be  put  with  the  four  blocks  to  make 
five  blocks? 

How  many  balls  do  I  move  on  this  wire  ?  "  Five 
balls."  How  many  balls  do  I  move  on  this  wire? 
"  Four  balls." 

Jane  may  put  five  shells  on  the  table;  John  four 
blocks;  and  George  three  buttons. 

How  many  more  shells  than  blocks  on  the  table? 
How  many  more  shells  than  buttons? 

NOTE. — Write  a  number  of  familiar  words  on  the  board — 
words  composed  of  three,  four,  and  five  letters — and  pointing  to 
different  words,  ask:  How  many  letters  in  this  word?  How 
many  in  this?  etc.  The  reading  chart  may  also  be  used,  and 
the  drill  should  be  continued  until  the  pupils  can  give  the  num- 
ber of  letters  in  each  word  without  counting. 


2.  How  many  balls  are  four  balls  and  one  ball?  One 
ball  and  four  balls? 

How  many  blocks  are  three  blocks  and  two  blocks? 
Two  blocks  and  three  blocks? 

Kate  may  put  three  shells  on  the  table,  and  Jane 
may  put  two  shells  with  Kate's.  How  many  shells  are 
on  the  table? 

How  many  shells  are  three  shells  and  two  shells? 

George  may  put  four  pebbles  on  the  table,  and 
Charles  may  put  enough  more  to  make  five  pebbles. 

How  many  pebbles  are  four  pebbles  and  one  pebble? 
One  pebble  and  four  pebbles? 

How  many  cents  are  three  cents  and  two  cents? 
Two  cents  and  three  cents? 


30  ORAL  LESSONS  IN  NUMBER. 

3.  Mary  may  hold  up  four  fingers  on  her  right  hand 
and  one  finger  on  her  left  hand.  How  many  fingers 
does  Mary  hold  up? 

Mary  may  put  down  the  one  finger  on  her  left  hand. 
How  many  fingers  does  she  now  hold  up? 

How  many  fingers  are  five  fingers  less  one  finger? 
Five  fingers  less  four  fingers? 

John  may  put  three  blocks  and  two  blocks  on  the 
table.  How  many  blocks  are  on  the  table? 

Clarence  may  take  away  two  of  the  blocks.  How 
many  blocks  are  left  on  the  table? 

How  many  blocks  are  five  blocks  less  two  blocks? 
Five  blocks  less  three  blocks? 

How  many  chairs  are  four  chairs  and  one  chair? 
Five  chairs  less  one  chair?  One  chair  and  four  chairs? 
Five  chairs  less  four  chairs? 

How  many  desks  are  three  desks  and  two  desks? 
Two  desks  and  three  desks?  Five  desks  less  two 
desks?  Five  desks  less  three  desks? 


4.  Five  acorns  are  four  acorns  and acorn,  or  one 

acorn  and  acorns. 

Five  acorns  are  three  acorns  and acorns,  or  two 

acorns  and  acorns. 

NOTE. — Separate  five  objects  of  different  kinds  into  two  groups, 
in  each  of  the  two  ways  indicated  above.  The  two  divisions  will 
give  these  results: 

Five  ohipoti  are  -i  4  °Mects  and  l  object,  or  1  object  and  4  objects. 
"\3  objects  and  2  objects,  or  2  objects  and  3  objects. 

John  may  divide  five  buttons  into  two  groups  in  two 
ways. 

NOTE. — After  the  first  division,  John  should  say,  "  Five  buttons 
are  four  buttons  and  one  button,  or  one  button  and  four  buttons." 
He  should  then  combine  the  two  groups  and  divide  again,  and 


FIRST-YEAR  COURSE.  31 

say,  "  Five  buttons  are  three  buttons  and  two  buttons,  or  two 
buttons  and  three  buttons." 

Lucy  may  divide  five  shells  into  two  groups  in  two 
ways. 

Five  desks  are  four  desks  and  -  -  desk,  or  one  desk 
and  desks. 

Five  desks  less  one  desk  are  desks;  five  desks 

less  four  desks  are  desk. 

Five  girls  are  three  girls  and  -  -  girls,  or  two  girls 
and  girls. 

Five  girls  less  two  girls  are  girls;  five  girls  less 

three  girls  are  girls. 


5.  Kate  may  hold  up  five  fingers,  and  Jane  may  hold 
up  three  fingers.  How  many  fingers  does  Kate  hold  up 
more  than  Jane? 

Harry  may  put  five  shells  and  four  pebbles  on  the 
table.  How  many  more  shells  than  pebbles  on  the 
table? 

How  many  pencils  in  my  right  hand?  "  Five  pen- 
cils." How  many  pencils  in  my  left  hand  ?  "  Three 
pencils."  How  many  more  pencils  in  my  right  hand 
than  in  my  left? 

NOTE. — Continue  the  comparison  of  unequal  groups  by  using 
books,  pebbles,  shells,  etc. 

How  many  lemons  are  two  lemons  and  three  lemons  ? 
Two  lemons  and  two  lemons?  One  lemon  and  four 
lemons?  Three  lemons  and  two  lemons?  Two  lemons 
and  one  lemon? 

How  many  oranges  are  five  oranges  less  one  orange? 
Four  oranges  less  two  oranges?  Five  oranges  less  two 
oranges?  Four  oranges  less  three  oranges?  Five 
oranges  less  three  oranges? 


32  ORAL  LESSONS  IN  NUMBER. 

6,  This  is  the  word  that  stands  for  five: 
This  is  the  figure  that  stands  for  five :  5. 
Make  the  figure  5  on  your  slate  five  times. 
Make  the  figures  1,  2,  3,  4,  5  on  your  slate 


LESSON  V. 
The  Number  6. 

1.  How  many  fingers  do  I  hold  up?  "Five  fingers." 
How  many  now  ?  "  Six  fingers."  Five  fingers  and  one 
finger  are  six  fingers. 

Hold  up  five  fingers.  How  many  more  fingers  must 
you  hold  up  to  make  six  fingers?  Hold  up  six  fingers. 

Albert  may  put  five  blocks  on  the  table,  and  Edward 
may  put  one  block  with  Albert's.  How  many  blocks 
are  now  on  the  table? 

Clara  may  put  four  buttons  on  the  table,  and  Agnes 
may  put  two  buttons  with  Clara's.  How  many  buttons 
are  on  the  table? 

How  many  balls  do  I  move  on  this  wire?  "Six 
balls."  How  many  on  this  wire?  "Five  balls." 

How  many  pencils  in  my  right  hand  ?  "  Four  pen- 
cils." How  many  in  my  left  hand?  "Six  pencils." 

How  many  books  do  I  hold  up?     "Six  books." 

How  many  letters  in  the  word  "Susan"?  In  the 
word  "Charles"? 

NOTE. — If  there  is  any  hesitancy  in  numbering  the  different 
groups  of  objects,  continue  the  drill,  using  various  objects,  and 
presenting  promiscuously  groups  of  three,  four,  five  and  six. 
Groups  of  the  figures  1,  2,  3,  4,  and  5,  respectively,  may  be  written 
on  the  board.  Pointing  to  the  different  groups,  ask :  How 
many  ones?  How  many  threes?  etc. 


FIRST- YEAR  COURSE.  33 

2.  How  many  balls  are  five  balls  and  one  ball?  One 
ball  and  five  balls? 

How  many  balls  are  four  balls  and  two  balls?  Two 
balls  and  four  balls? 

How  many  balls  are  three  balls  and  three  balls? 

Willie  may  put  four  blocks  on  the  table,  and  Alice 
may  put  two  blocks  with  Willie's.  How  many  blocks 
are  on  the  table? 

How  many  blocks  are  four  blocks  and  two  blocks? 
Two  blocks  and  four  blocks? 

George  may  hold  up  three  fingers  on  his  right  hand, 
and  three  fingers  on  his  left  hand.  How  many  fingers 
does  George  hold  up? 

Hold  up  two  fingers  and  four  fingers.  How  many 
fingers  do  you  hold  up? 

How  many  desks  are  five  desks  and  one  desk?  One 
desk  and  five  desks? 

How  many  desks  are  four  desks  and  two  desks? 
Two  desks  and  four  desks?  Three  desks  and  three 
desks? 


3.  Charles  may  put  four  shells  and  two  shells  on  the 
table.  How  many  shells  are  on  the  table? 

Clara  may  take  away  two  of  the  shells.  How  many 
shells  are  left  on  the  table? 

How  many  shells  are  six  shells  less  two  shells?  Six 
shells  less  four  shells? 

Hold  up  six  fingers,  three  on  each  hand ;  put  down 
three  fingers.  How  many  fingers  are  left? 

How  many  fingers  are  six  fingers  less  three  fingers? 

How  many  oranges  are  six  oranges  less  one  orange? 
Six  oranges  less  five  oranges?  Six  oranges  less  two 
oranges?  Six  oranges  less  four  oranges?  Six  oranges 
less  three  oranges? 


34  ORAL  LESSONS  IN  NUMBER. 

How  many  tops  are  five  tops  and  one  top?  Six  tops 
less  one  top?  One  top  and  five  tops?  Six  tops  less 
five  tops? 

How  many  tops  are  four  tops  and  two  tops?  Six 
tops  less  two  tops?  Two  tops  and  four  tops?  Six  tops 
less  four  tops? 

How  many  tops  are  three  tops  and  three  tops?  Six 
tops  less  three  tops? 


4.  Here  are  six  shells;  let  us  see  in  how  many  ways 
we  can  separate  them  into  two  groups: 

Six  shells  are  five  shells  and  shell,  or  one  shell 

and  shells. 

Six  shells  are  four  shells  and  shells,  or  two 

shells  and  shells. 

Six  shells  are  three  shells  and  shells. 

NOTE. — Take  six  shells  and  divide  them  successively  into  two 
groups  as  above  indicated,  and  have  the  pupils  say :  "  Six  shells 
are  five  shells  and  one  shell,  or  one  shell  and  five  shells,"  etc. 

Mary  may  take  six  spools  and  separate  them  into  two 
groups;  Kate,  six  shells  and  separate  them  into  two 
other  groups;  and  Agnes,  six  buttons  and  separate 
them  into  two  other  groups. 

John  may  take  six  blocks  and  divide  them  into  two 
groups  in  three  ways. 


NOTE. — The  three  divisions  will  give  these  results: 

{5  blocks  and  1  block,  or  1  block  and  5  blocks. 
4  blocks  and  2  blocks,  or  2  blocks  and  4  blocks. 
3  blocks  and  3  blocks. 


Kate  may  divide  six  shells  into  two  equal  groups. 
How  many  shells  in  each  group?  How  many  three 
shells  in  six  shells? 


FIRST-YEAR   COURSE. 


35 


Willie  may  divide  six  pencils  into  three  equal  groups. 
How  many  pencils  in  each  group?  How  many  two 
pencils  in  six  pencils? 


5.  Six  apples  are  five  apples  and  apple,  or  one 

apple  and  -  -  apples. 

Six  apples  less  one  apple  are apples;  six  apples 

less  five  apples  are  apple. 

Six  lemons  are  four  lemons  and lemons,  or  two 


lemons  and 


lemons. 


Six  lemons  less  two  lemons  are  lemons ;  six 

lemons  less  four  lemons  are  lemons. 

Six  lemons  are  three  lemons  and  lemons;  six 

lemons  less  three  lemons  are  lemons. 


6.  How  many  men  are : 
Two  men  and  two  men? 
Three  men  and  two  men? 
Four  men  and  two  men? 
One  man  and  three  men? 
Two  men  and  three  men  ? 
Three  men  and  three  men  ? 
One  man  and  four  men? 
Two  men  and  four  men? 
One  man  and  five  men? 


Four  men  less  two  men  ? 
Five  men  less  two  men? 
Six  men  less  two  men? 
Four  men  less  three  men  ? 
Five  men  less  three  men  ? 
Six  men  less  three  men? 
Five  men  less  four  men? 
Six  men  less  four  men? 
Six  men  less  five  men? 


NOTE.— Let  these  exercises  be  first  recited  across  the  page ;  as, 
"  Two  men  and  two  men  are  four  men ;  four  men  less  two  men 
are  two  men."  Then  have  the  exercises  in  addition  and  subtrac- 
tion recited  separately.  Also  give  the  exercises  in  a  miscellaneous 
manner. 


36  OEAL  LESSONS  IN  NUMBER. 

7.  Maud  may  hold  up  six  fingers,  and  Mary  may 
hold  up  four  fingers.  How  many  fingers  does  Maud 
hold  up  more  than  Mary  ? 

There  are  six  boys  and  four  girls  in  a  class.  How 
many  more  boys  than  girls  in  the  class? 

John  found  six  eggs  in  a  nest,  and  Henry  found 
five  eggs  in  another  nest.  How  many  more  eggs  did 
John  find  than  Henry? 

I  have  six  pencils  in  my  right  hand  and  three  pen- 
cils in  ray  left  hand.  How  many  more  pencils  in  my 
right  hand  than  in  my  left? 

Clara  found  four  eggs  in  one  nest  and  two  eggs  in 
another  nest.  How  many  eggs  did  she  find? 

A  boy  bought  six  peaches  and  gave  three  of  them  to 
his  sister.  How  many  peaches  had  he  left? 

A  man  paid  four  dollars  for  a  hat  and  two  dollars 
for  a  cap.  How  many  dollars  did  he  pay  for  both? 

Harry  earned  six  cents  and  then  gave  two  cents  for  a 
pencil.  How  many  cents  had  he  left? 

Willie  is  six  years  old,  and  Fanny  is  four  years  old. 
How  many  years  older  than  Fanny  is  Willie? 


8.  This  is  the  word  that  stands  for  six : 
This  is  the  figure  that  stands  for  six :  6. 
Make  the  figure  6  on  your  slate  six  times. 
Make  the  figures  1,  2,  3,  4,  5,  6  on  your  slate. 


LESSON  VI. 

The  Number  7. 


1.  How  many  fingers  do  I  hold  up  on  my  right 
hand  ?  "  Four  fingers."  How  many  on  my  left  hand  ? 
"Three  fingers."  How  many  on  both  hands? 


FIRST- YEAR  COURSE.  37 

Hold  up  six  fingers.  How  many  fingers  more  will 
make  seven  fingers?  Hold  up  seven  fingers. 

Jane  may  put  six  shells  on  the  table ;  Kate  may  put 
one  shell  with  Jane's.  How  many  shells  are  now  on 
the  table? 

How  many  shells  do  I  put  on  the  table?  "  Four 
shells."  How  many  now?  "Seven  shells." 

NOTE. — The  teacher  should  put  down  four  shells,  six  shells, 
seven  shells,  five  shells,  etc.,  and  continue  until  the  pupils  can 
give  the  number  of  shells  instantly.  Various  other  objects  may 
be  used.  The  pupils  may  also  be  required  to  give  the  number  of 
panes  of  glass  in  the  window,  books  on  the  desk,  desks  in  a  row, 
etc. 


2.  How  many  tops  are  six  tops  and  one  top?  Five 
tops  and  two  tops? 

How  many  tops  are  four  tops  and  three  tops? 

Susan  may  put  four  shells  on  the  table,  and  Mary 
may  put  three  shells  near  Susan's.  How  many  shells 
are  on  the  table?  (Sliding  the  groups  together.) 

How  many  shells  are  four  shells  and  three  shells? 

James  may  hold  up  five  counters  and  Willie  may 
hold  up  two  counters.  How  many  counters  do  both 
hold  up? 

How  many  counters  are  five  counters  and  two 
counters  ? 

How  many  chairs  are  six  chairs  and  one  chair? 
Five  chairs  and  two  chairs?  Four  chairs  and  three 
chairs  ? 

How  many  pencils  are  three  pencils  and  four  pen- 
cils? Two  pencils  and  five  pencils?  One  pencil  and 
six  pencils? 

How  many  buttons  do  I  put  on  the  table  ?  "  Five 
buttons."  How  many  more?  "Two  buttons."  How 
many  buttons  are  now  on  the  table? 


38  ORAL  LESSONS  IN  NUMBER. 

Harry  may  take  up  two  buttons.  How  many  buttons 
are  left  on  the  table? 

How  many  buttons  are  seven  buttons  less  two  but- 
tons? 


3,  Jane  may  put  four  spools  and  three  spools  on  the 
table.  How  many  spools  are  on  the  table? 

Mary  may  take  away  three  of  the  spools.  How 
many  spools  are  left  on  the  table? 

How  many  spools  are  seven  spools  less  three  spools? 

Herbert  may  put  seven  marbles  in  my  hand,  and 
Merrill  may  take  four  of  them.  How  many  marbles 
are  left  in  my  hand?  • 

How  many  marbles  are  seven  marbles  less  four  mar- 
bles? 

How  many  balls  are  seven  balls  less  five  balls? 
Seven  balls  less  two  balls? 

How  many  shells  are  six  shells  less  four  shells? 
Five  shells  less  three  shells?  Seven  shells  less  three 
shells  ? 

How  many  boys  are  six  boys  and  one  boy?  Seven 
boys  less  one  boy?  One  boy  and  six  boys?  Seven 
boys  less  six  boys? 

How  many  boys  are  five  boys  and  two  boys?  Seven 
boys  less  two  boys?  Two  boys  and  five  boys?  Seven 
boys  less  five  boys? 

How  many  girls  are  four  girls  and  three  girls? 
Seven  girls  less  three  girls  ?  Three  girls  and  four  girls  ? 
Seven  girls  less  four  girls? 


4,  Seven  rings  are  six  rings  and  ring>  °r  °ne 

ring  and  rings. 

Seven  rings  are  five  rings  and rings,  or  two  rings 

and  rings. 


FIRST- YEAR   COURSE. 


39 


Seven  rings  are  four  rings  and  rings,  or  three 

rings  and  rings. 

NOTE. — Take  the  seven  objects  and  divide  them  successively 
into  the  two  groups  above  indicated,  and  have  the  pupils  give  the 
number  of  objects  in  each  of  the  two  groups. 

John  may  take  seven  blocks  and  divide  them  into 
two  groups  ;  George,  seven  buttons  and  divide  them  into 
two  other  groups;  and  Willie,  seven  shells  and  divide 
them  into  two  other  groups.  Each  boy  may  tell  us 
the  result  of  his  division. 

Albert  may  take  seven  shells  and  divide  them  into 
two  groups  in  as  many  ways  as  he  can. 


NOTE. — These  divisions  will  give  the  following  results : 

!6  shells  and  1  shell,  or  1  shell  and  6  shells. 
5  shells  and  2  shells,  or  2  shells  and  5  shells. 
4  shells  and  3  shells,  or  3  shells  and  4  shells. 


Susan  may  take  seven  buttons  and  divide  them  into 
two  groups  in  three  ways. 


5.  Seven  chairs  are  six  chairs  and  - 
chair  and  chairs. 

Seven  chairs  less  one  chair  are  — 
chairs  less  six  chairs  are  chair. 

Seven  chairs  are  five  chairs  and  — 
chairs  and  chairs. 

Seven  chairs  less  two  chairs  are  — 
chairs  less  five  chairs  are  chairs. 

Seven  chairs  are  four  chairs  and  

chairs  and chairs. 

Seven  chairs  less  three  chairs  are  — 
chairs  less  four  chairs  are  chairs. 

Seven  cents  are  five  cents  and  — 
cents  less  five  cents  are  cents. 


chair,  or  one 


chairs ;  seven 
chairs,  or  two 
chairs ;  seven 


chairs,  or  three 
-  chairs;  seven 


cents ;    seven 


40  ORAL  LESSONS  IN  NUMBER. 

6.  How  many  lines  are: 

Two  lines  and  two  lines  ?  Four  lines  less  two  lines  ? 

Three  lines  and  two  lines  ?  Five  lines  less  two  lines  ? 

Four  lines  and  two  lines?  Six  lines  less  two  lines? 

Five  lines  and  two  lines  ?  Seven  lines  less  two  lines  ? 

One  line  and  three  lines  ?  Four  lines  less  three  lines  ? 

Two  lines  and  three  lines  ?  Five  lines  less  three  lines  ? 

Three  lines  and  three  lines  ?  Six  lines  less  three  lines  ? 

Four  lines  and  three  lines  ?  Seven  lines  less  three  lines? 
How  many  dots  are: 

One  dot  and  four  dots  ?  Five  dots  less  four  dots  ? 

Two  dots  and  four  dots?  Six  dots  less  four  dots? 

Three  dots  and  four  dots  ?  Seven  dots  less  four  dots  ? 

One  dot  and  five  dots  ?  Six  dots  less  five  dots  ? 

Two  dots  and  five  dote  ?  Seven  dots  less  five  dots  ? 

NOTE.— For  directions,  see  Lesson  V,  page  35,  note. 


7.  Willie  may  hold  up  seven  fingers,  and  Harry  five 
fingers.  How  many  more  fingers  does  Willie  hold  up 
than  Harry? 

James  may  put  seven  pebbles  on  the  table,  and 
George  four  pebbles.  How  many  pebbles  in  James's 
group  more  than  in  George's?  George  may  put  enough 
pebbles  in  his  group  to  make  it  equal  to  James's. 

I  put  seven  books  in  one  pile  and  five  books  in 
another  pile.  How  many  books  must  I  add  to  the 
second  pile  to  make  it  equal  in  number  to  the  first 
pile? 

I  give  Kate  six  cents  and  Mary  four  cents.  How 
many  more  cents  have  I  given  to  Kate  than  to  Mary? 

How  many  cents  in  seven  cents  more  than  in  five 
cents?  In  seven  cents  more  than  in  four  cents? 


FIRST-YEAR  COURSE.  41 

8.  Samuel  earned  seven  cents,  and  paid  five  cents  for 
a  book.  How  many  cents  had  he  left? 

Sarah  picked  four  roses  from  one  bush  and  three 
roses  from  another  bush.  How  many  roses  did  she 
pick?  If  she  should  give  three  roses  to  her  sister,  how 
many  would  she  have  left? 

A  farmer  has  five  horses  in  one  field  and  two  horses 
in  another  field.  How  many  horses  in  both  fields? 

Jane  is  seven  years  old,  and  her  sister  is  three  years 
old.  How  many  years  older  than  her  sister  is  Jane? 


9.  This  is  the  word  that  stands  for  seven : 
This  is  the  figure  that  stands  for  seven :  7. 
Make  the  figure  7  on  your  slate  seven  times. 
Make  the  figures  1,  2,  3,  4,  5,  6,  7  on  your  slate. 


LESSON  VII. 
The  Number  8. 

1.  How  many  fingers  do  I  hold  up  ?    "  Seven  fingers. 
How  many  fingers  do  I  now  hold  up?     "One  finger." 
Seven  fingers  and  one  finger  are  eight  fingers. 

Hold  up  seven  fingers.  How  many  fingers  more  will 
make  eight  fingers?  Hold  up  eight  fingers. 

Frank  may  put  seven  blocks  on  the  table,  and  Willie 
may  put  one  block  with  Frank's.  How.  many  blocks 
are  now  on  the  table? 

How  many  books  do  I  hold  up?  "Eight  books." 
How  many  now?  (Taking  away  one  book.) 

How  many  fingers  do  I  hold  up?  "Six  fingers." 
How  many  now  ?  "  Eight  fingers."  How  many  now  ? 
"Seven  fingers." 

O.  L.-4. 


42  ORAL  LESSONS  IN  NUMBER. 

How  many  pencils  do  I  hold  up?  "Six  pencils." 
How  many  now  ?  u  Seven  pencils."  How  many  now  ? 
"Eight  pencils." 

NOTE. — If  there  is  any  hesitancy  in  numbering  the  different 
groups  of  objects,  continue  the  drill,  using  various  objects,  and 
presenting  promiscuously  groups  of  four,  five,  six,  seven,  and 
and  eight.  Write  groups  of  the  figures  1,  2,  3,  4,  etc.,  on  the 
board,  and,  pointing  to  the  different  groups,  ask :  How  many 
twos  in  this  group?  How  many  fours  in  this?  etc. 


2.  How  many  balls  are  seven  balls  and  one  ball? 
Six  balls  and  two  balls?  Two  balls  and  six  balls? 

NOTE. — Present  the  groups  of  balls  separately,  and  then  slide 
them  together  as  the  question  is  asked. 

George  may  put  six  shells  on  the  table,  and  Charles 
may  put  two  shells  with  George's.  How  many  shells 
are  on  the  table? 

How  many  shells  are  six  shells  and  two  shells? 
Two  shells  and  six  shells? 

Here  are  five  balls,  and  here  are  three  balls.  How 
many  balls  do  I  slide  together? 

Jane  may  hand  me  five  shells,  and  Lucy  may  hand 
me  three  shells.  How  many  shells  have  I? 

How  many  shells  are  five  shells  and  three  shells? 
Three  shells  and  five  shells? 

How  many  fingers  do  I  hold  up  on  my  right  hand? 
"Four  fingers."  How  many  on  my  left  hand?  "Four 
fingers."  How  many  on  both  hands? 

Clara  may  hold  up  four  fingers,  and  Helen  may  hold 
up  four  fingers.  How  many  fingers  do  both  hold  up? 

How  many  fingers  are  four  fingers  and  four  fingers? 

How  many  birds  are  seven  birds  and  one  bird?  Six 
birds  and  two  birds?  Five  birds  and  three  birds? 
Four  birds  and  four  birds? 


FIRST- YEAR  COURSE.  43 

How  many  boys  are  one  boy  and  seven  boys?  Two 

boys  and  six  boys?    Three  boys  and  five  boys?  Four 
boys  and  four  boys? 


3.  How  many  blocks  do  I  put  on  the  table  ?  "  Six 
blocks."  How  many  blocks  do  I  now  put  near  them  ? 
"  Two  blocks."  How  many  blocks  are  on  the  table  ? 

I  now  take  away  the  two  blocks.  How  many  blocks 
are  left  on  the  table? 

Eight  blocks  less  two  blocks  are  how  many  blocks? 

I  again  put  the  two  blocks  with  the  six  blocks. 
How  many  blocks  are  on  the  table?  I  now  take  away 
six  blocks.  How  many  blocks  are  left  on  the  table? 

Eight  blocks   less  six  blocks  are  how  many  blocks? 

Susan  may  put  eight  shells  on  the  table,  and  Alice 
may  take  away  three  of  the  shells.  How  many  shells 
are  left  on  the  table? 

How  many  shells  are  eight  shells  less  three  shells? 
Eight  shells  less  five  shells? 

Harry  may  take  eight  shells,  and  hand  four  of  them 
to  Thomas.  How  many  shells  has  Harry  left? 

How  many  shells  are  eight  shells  less  four  shells? 

How  many  plums  are  seven  plums  and  one  plum  ? 
Eight  plums  less  one  plum? 

How  many  pears  are  six  pears  and  two  pears?  Eight 
pears  less  two  pears? 

How  many  peaches  are  five  peaches  and  three 
peaches?  Eight  peaches  less  three  peaches? 

How  many  oranges  are  four  oranges  and  four 
oranges?  Eight  oranges  less  four  oranges? 

How  many  cents  are  five  cents  and  three  cents? 
Eight  cents  less  three  cents? 

How  many  cents  are  four  cents  and  four  cents? 
Eight  cents  less  four  cents? 


44  ORAL  LESSONS  IN  NUMBER. 

4.  Let  us  take  eight  blocks  and  see  in  how  many 
ways  we  can  divide  them  into  two  groups : 

(1)  Eight  blocks  are  seven  blocks  and block,  or 

one  block  and  blocks. 

(2)  Eight  blocks  are  six  blocks  and  blocks,  or 

two  blocks  and  blocks. 

(3)  Eight  blocks  are  five  blocks  and  blocks,  or 

three  blocks  and  blocks. 

(4)  Eight  blocks  are  four  blocks  and  blocks. 

Here  are  eight  pretty  acorn  cups.     Mary  may  divide 

them   into   two  groups   in   four  ways,   and   after   each 
division  she  may  tell  me  the  result: 

NOTE. — The  four  divisions  will  give  these  results: 

f  7  cups  and  1  cup,  or  1  cup  and  7  cups. 

-r,.  i  4.  6  cups  and  2  cups,  or  2  cups  and  6  cups. 

Eight  cups  are  :  j  5  ^  and  g  c^  Qr  g  ^  Rnd  5  ^ 

[.4  cups  and  4  cups. 

I  now  put  on  the  table  eight  blocks,  eight  shells, 
eight  pebbles,  and  eight  buttons.  Harry  may  divide 
the  blocks  into  two  groups;  Charles,  the  shells  into  two 
other  groups ;  George,  the  pebbles  into  two  other  groups  ; 
and  Frank,  the  buttons  into  two  other  groups.  Each 
boy  may  now  tell  me  the  result  of  his  division. 

Agnes  may  take  eight  shells  and  divide  them  into 
two  groups  in  as  many  ways  as  she  can,  telling  me  the 
result  after  each  division. 

Kate  may  take  eight  pencils  and  divide  them  into 
two  equal  groups.  How  many  pencils  in  each  group? 

How  many  four  pencils  make  eight  pencils? 

Jane  may  take  eight  shells  and  divide  them  into  four 
equal  groups.  How  many  shells  in  each  group? 

How  many  two  shells  make  eight  shells? 


5.  Eight  birds  are  seven  birds  and  bird,  or  one 

bird  and  birds. 


FIRST-YEAR  COURSE.  45 

Eight  birds  less  one  bird  are  -  -  birds;  eight  birds 
less  seven  birds  are  bird. 

Eight  men  are  six  men  and  — —  men,  or  two  men 
and men. 

Eight  men  less  two  men  are  -  -  men ;  eight  men 
less  six  men  are  men. 

Eight  boys  are  five  boys  and boys,  or  three  boys 

and boys. 

Eight  boys  less  three  boys  are boys ;  eight  boys 

less  five  boys  are  boys. 

Eight  horses  are  four  horses  and  horses. 

Eight  horses  less  four  horses  are  horses. 


6.  Kate  may  hold  *up  six  fingers,  and  Lucy  eight 
fingers.  How  many  more  fingers  does  Lucy  hold  up 
than  Kate? 

I  have  seven  pencils  in  my  right  hand  and  four  pen- 
cils in  my  left  hand.  How  many  more  pencils  in  my 
right  hand  than  in  my  left? 

Kate  picked  eight  roses  and  five  tulips.  How  many 
more  roses  than  tulips  did  she  pick? 

Clara  has  eight  cents,  and  Ruth  three  cents.  How 
many  cents  has  Clara  more  than  Ruth? 

How  many  cents  must  I  give  to  Ruth  that  she  may 
have  as  many  as  Clara? 

How  many  cents  must  be  added  to  six  cents  to  make 
eight  cents? 

There  are  eight  sheep  in  one  field,  and  five  sheep  in 
another  field.  How  many  more  sheep  in  the  first  field 
than  in  the  second? 

Mary  found  six  eggs  in  one  nest  and  eight  eggs  in 
another  nest.  How  many  eggs  in  the  first  nest  less 
than  in  the  second?  How  many  eggs  in  the  second 
nest  more  than  in  the  first? 


46  ORAL  LESSONS  IN  NUMBER. 

7.  There  are  four  desks  in  one  row,  and  four  desks  in 
another  row.  How  many  desks  in  both  rows? 

Clarence  bought  eight  peaches  and  gave  four  of  them 
to  his  sister.  How  many  peaches  had  he  left? 

There  are  four  boys  and  three  girls  in  a  class.  How 
many  pupils  in  the  class? 

John  gave  three  cents  for  a  pencil  and  five  cents  for 
paper.  How  many  cents  did  he  pay  for  pencil  and  paper? 

There  are  two  men  plowing  in  one  field,  and  five  men 
in  another  field.  How  many  men  in  both  fields? 

There  were  eight  birds  in  a  tree,  and  five  of  them 
flew  away.  How  many  birds  were  left  in  the  tree? 

Mary  wrote  eight  words  on  her  slate  and  then  rubbed 
out  three  words.  How  many  words  were  left? 

Susan  picked  eight  tulips  and  gave  two  of  them  to 
Kate.  How  many  tulips  had  Susan  left? 


8.  This  is  the  word  that  stands  for  eight:  $*/f/ 
This  is  the  figure  that  stands  for  eight :  $. 
Make  the  figure  8  on  your  slate  eight  times. 
Make  the   figures   1,  2,  3,  4,  5,  6,  7,  8  on  your  slate. 


REVIEW    EXERCISES. 

How  many  cents  are: 

Two  cents  and  two  cents  ?  Four  cents  less  two  cents  ? 

Three  cents  and  two  cents?  Five  cents  less  two  cents? 

Four  cents  and  two  cents  ?  Six  cents  less  two  cents  ? 

Five  cents  and  two  cents  ?  Seven  cents  less  two  cents  ? 

Six  cents  and  two  cents  ?  Eight  cents  less  two  cents  ? 

Two  cents  and  three  cents  ?  Five  cents  less  three  cents  ? 

Three  cents  and  three  cents  ?  Six  cents  less  three  cents  ? 

Four  cents  and  three  cents  ?  Seven  cents  less  three  cents  ? 

Five  cents  and  three  cents  ?  Eight  cents  less  three  cents  ? 


FIRST-YEAR  COURSE.  47 

How  many  figs  are: 

One  fig  and  four  figs  ?  Five  figs  less  four  figs  ? 

Two  figs  and  four  figs  ?  Six  figs  less  four  figs  ? 

Three  figs  and  four  figs  ?  Seven  figs  less  four  figs  ? 

Four  figs  and  four  figs  ?  Eight  figs  less  four  figs  ? 

One  fig  and  five  figs  ?  Six  figs  less  five  figs  ? 

Two  figs  and  five  figs  ?  Seven  figs  less  five  figs  ? 

Three  figs  and  five  figs  ?  Eight  figs  less  five  figs  ? 

Two  figs  and  six  figs  ?  Eight  figs  less  six  figs  ? 


LESSON  VIII. 

The  Number  9. 

1.  How  many  balls  do  I  move  on  this  wire  ?  "  Eight 
balls."  How  many  do  I  now  move  towards  the  eight 
balls?  "One  ball." 

Eight  balls  and  one  ball  (sliding  them  together)  are 
nine  balls.  How  many  balls  do  I  now  move  ?  "  Nine 
balls." 

How  many  books  have  I  put  on  the  table?  "Nine 
books."  I  take  away  one  book  :  how  many  books  are 
left? 

How  many  books  must  I  add  to  eight  books  to  make 
nine  books? 

Edward  may  put  eight  blocks  on  the  table,  and 
Charles  may  put  one  block  with  Edward's.  How  many 
blocks  are  on  the  table? 

Jane  may  put  nine  shells  on  the  table;  Susan,  nine 
rings ;  and  Alice,  nine  counters. 

I  will  write  several  groups  of  the  figure  5  on  the 
board.  How  many  5's  in  this  group  ?  "  Seven  5's." 
How  many  5's  in  this?  "Nine  5's."  How  many  in 
this?  "Six  5's."  How  many  in  this?  "  Eight  5's." 

NOTE. — Write  groups  of  other  figures  on  the  board,  and  con- 
tinue the  drill  until  they  are  numbered  without  hesitation. 


48  ORAL  LESSONS  IN  NUMBER. 

2.  How  many  balls  are  eight  balls  and  one  ball? 
Seven  balls  and  two  balls?  Six  balls  are  three  balls? 
Five  balls  and  four  balls? 

NOTE. — First  present  the  groups  separately,  and  then  slide 
them  together  as  the  question  is  asked. 

Jane  may  put  seven  shells  on  the  table,  and  Susan 
may  put  two  shells  near  Jane's.  How  many  shells  on 
the  table? 

Seven  shells  and  two  shells  are  how  many  shells? 

NOTE. — Slide  the  groups  together  as  the  question  is  asked. 

Kate  may  put  six  rings  on  the  table,  and  Mary  may 
put  three  rings  near  Kate's.  How  many  rings  are  on 
the  table? 

Six  rings  and  three  rings  are  how  many  rings? 

Samuel  may  put  five  pebbles  in  my  right  hand  and 
four  pebbles  in  my  left  hand.  How  many  pebbles  in 
both  of  my  hands? 

How  many  pebbles  are  five  pebbles  and  four  pebbles? 

How  many  chairs  are  eight  chairs  and  one  chair? 
Seven  chairs  and  two  chairs?  Six  chairs  and  three 
chairs?  Five  chairs  and  four  chairs? 


3.  Willie  may  put  three  pebbles  on  the  table ;  Susan, 
three  shells;  and  Charles,  three  cents.  How  many 
objects  on  the  table? 

Willie  may  take  away  his  three  pebbles.  How  many 
objects  are  left? 

Susan  may  next  take  away  her  three  shells.  What 
is  now  left  on  the  table? 

How  many  balls  have  I  moved  on  the  wire?  "Nine 
balls."  How  many  balls  do  I  take  away?  "Two 
balls."  How  many  balls  are  left?. 


FIRST- YEAR  COURSE.  49 

How  many  balls  are  nine  balls  less  two  balls? 

Harry  may  put  nine  shells  on  the  table,  and  Charles 
may  take  away  three  shells.  How  many  shells  are 
left? 

Nine  shells  less  three  shells  are  how  many  shells? 

I  give  George  nine  buttons,  and  he  may  give  four 
buttons  to  James.  How  many  buttons  has  George  left? 

Nine  buttons  less  four  buttons  are  how  many  but- 
tons? 

How  many  pens  are  eight  pens  and  one  pen?  Nine 
pens  less  one  pen?  One  pen  and  eight  pens?  Nine 
pens  less  eight  pens? 

How  many  apples  are  seven  apples  and  two  apples? 
Nine  apples  less  two  apples?  Two  apples  and  seven 
apples?  Nine  apples  less  seven  apples? 

How  many  lemons  are  six  lemons  and  three  lemons? 
Nine  lemons  less  three  lemons?  Three  lemons  and  six 
lemons?  Nine  lemons  less  six  lemons? 

How  many  figs  are  five  figs  and  four  figs?  Nine  figs 
less  four  figs?  Four  figs  and  five  figs?  Nine  figs  less 
five  figs? 


4.  I  will  take  nine  pencils,  and  see  in  how  many 
ways  I  can  divide  them  into  two  groups : 

(1)  Nine  pencils  are  eight  pencils  and pencil,  or 

one  pencil  and  pencils. 

(2)  Nine  pencils  are  seven  pencils  and  pencils, 

or  two  pencils  and  pencils. 

(3)  Nine  pencils  are  six  pencils  and  pencils,  or 

three  pencils  and  pencils. 

(4)  Nine  pencils  are  five  pencils  and pencils,  or 

four  pencils  and  pencils. 

I  now  put  on  the  table  nine  blocks,  nine  shells,  nine 
pebbles,  and  nine  buttons.  Albert  may  divide  the 
blocks  into  two  groups  ;  Edward,  the  shells  into  two 

O.  L.-5. 


50  ORAL  LESSONS  IN  NUMBER. 

other  groups;  William,  the  pebbles  into  two  other 
groups;  and  Calvin,  the  buttons  into  two  other  groups. 
Each  boy  may  tell  me  the  result. 

Here  are  nine  pretty  acorn  cups.  Clara  may  divide 
them  into  two  groups  in  four  ways,  and  after  each  divi- 
sion she  may  tell  me  the  result. 

NOTE. — The  four  divisions  will  give  these  results: 

f  8  cups  and  1  cup,  or  1  cup  and  8  cups. 

XT-   „  I  7  cups  and  2  cups,  or  2  cups  and  7  cups. 

Nine  cups  are :    ?  ^g  and  g  ^  Qr  3  ^  and  6  ^ 

I  5  cups  and  4  cups,  or  4  cups  and  5  cups. 

Harry  may  take  nine  shells  and  divide  them  into 
two  groups  in  as  many  ways  as  he  can. 

Charles  may  take  nine  blocks  and  divide  them  into 
three  equal  groups.  How  many  blocks  in  each  group? 

How  many  three  blocks  in  nine  blocks? 


5.  Nine  plums  are  eight  plums  and plum,  or  one 

plum  and  plums. 

Nine  plums  less  one  plum  are  plums;  nine 

plums  less  eight  plums  are  plum. 

Nine  pears  are  seven  pears  and  pears,  or  two 

pears  and  pears. 

Nine  pears  less  two  pears  are  pears;  nine  pears 

less  seven  pears  are  pears. 

Nine  peaches  are  six  peaches  and  peaches,  or 

three  peaches  and  peaches. 

Nine  peaches  less  three  peaches  are  peaches; 

nine  peaches  less  six  peaches  are  peaches. 

Nine  figs  are  five  figs  and  figs,  or  four  figs  and 

figs. 

Nine  figs  less  four  figs  are figs ;  nine  figs  less  five 

figs  are figs. 


FIRST-YEAR  COURSE.  51 

6.  I  have  nine  pencils  in  my  right  hand  and  five 
pencils  in  my  left  hand.  How  many  pencils  in  my 
right  hand  more  than  in  my  left? 

Jane  may  put  nine  shells  on  the  table,  and  Kate  six 
shells.  How  many  shells  in  Jane's  group  more  than  in 
Kate's?  How  many  shells  must  I  take  from  Kate's  to 
make  the  two  groups  equal?  How  many  shells  must  I 
add  to  Jane's  to  make  the  two  groups  equal? 

I  give  John  eight  pencils  and  Harry  five  pencils. 
How  many  pencils  has  John  more  than  Harry?  How 
many  pencils  must  I  give  to  Harry  to  make  his  pencils 
equal  John's  in  number? 

Jane  has  nine  cents,  and  Susan  seven  cents.  How 
many  cents  has  Jane  more  than  Susan? 


7.  Charles  gave  eight  cents  for  raisins  and  five  cents 
for  an  orange.  How  many  more  cents  did  he  pay  for 
the  raisins  than  for  the  orange? 

Orvil  caught  eight  fishes  and  sold  three  of  them. 
How  many  fishes  had  he  left? 

There  are  five  trees  in  one  row,  and  four  trees  in 
another  row.  How  many  trees  in  both  rows? 

Clarence  has  six  marbles  in  his  right  hand  and  two 
marbles  in  his  left  hand.  How  many  marbles  in  both 
of  his  hands? 

George  is  eight  years  old,  and  Hiram  is  six  years  old. 
How  many  years  older  is  George  than  Hiram? 


8.  This  is  the  word  that  stands  for  nine: 

This  is  the  figure  that  stands  for  nine :  9. 

Make  the  figure  9  on  your  slate  nine  times. 

Make  the  figures  1,  2,  3,  4,  5,  6,  7,  8,  9  on  your  slate. 


52  ORAL  LESSONS  IN  NUMBER. 

REVIEW    EXERCISES. 

How  many  cents  are  : 

Four  cents  and  two  cents  ?    Six  cents  less  two  cents  ? 
Five  cents  and  two  cents  ?     Seven  cents  less  two  cents  ? 
Six  cents  and  two  cents  ?       Eight  cents  less  two  cents  ? 
Seven  cents  and  two  cents  ?  Nine  cents  less  two  cents  ? 

Three  cents  and  three  cents  ?  Six  cents  less  three  cents  ? 
Four  cents  and  three  cents  ?  Seven  cents  less  three  cents  ? 
Five  cents  and  three  cents  ?    Eight  cents  less  three  cents  ? 
Six  cents  and  three  cents  ?     Nine  cents  less  three  cents  ? 

Two  pens  and  four  pens  ?  Six  pens  less  four  pens  ? 

Three  pens  and  four  pens  ?  Seven  pens  less  four  pens  ? 

Four  pens  and  four  pens  ?  Eight  pens  less  four  pens  ? 

Five  pens  and  four  pens  ?  Nine  pens  less  four  pens  ? 

One  pen  and  five  pens  ?  Six  pens  less  five  pens  ? 

Two  pens  and  five  pens  ?  Seven  pens  less  five  pens  ? 

Three  pens  and  five  pens  ?  Eight  pens  less  five  pens  ? 

Four  pens  and  five  pens  ?  Nine  pens  less  five  pens  ? 

One  pear  and  six  pears  ?  Seven  pears  less  six  pears  ? 
Two  pears  and  six  pears  ?  Eight  pears  less  six  pears  ? 
Three  pears  and  six  pears  ?  Nine  pears  less  six  pears  ? 

Two  pears  and  seven  pears  ?   Nine  pears  less  seven  pears  ? 
Four  pears  and  five  pears  ?     Nine  pears  less  five  pears  ? 
Six  pears  and  three  pears  ?      Nine  pears  less  three  pears  ? 


LESSON  IX. 

The  Number  10. 

1.  How  many  balls  do  I  move  on  the  wire?  "Nine 
balls."  How  many  balls  do  I  now  move?  "One  ball." 
Nine  balls  and  one  ball  are  ten  balls.  ' 


FIRST- YEAR  COURSE.  53 

How  many  balls  on  this  wire?  "Ten  balls."  How 
many  balls  on  this  wire?  "Ten  balls." 

How  many  fingers  do  I  hold  up  ?  "  Eight  fingers." 
How  many  thumbs?  "Two  thumbs."  How  many 
fingers  and  thumbs?  "Ten  fingers  and  thumbs." 

Hold  up  ten  fingers  and  thumbs. 

Here  is  a  pair  of  gloves.  How  many  fingers  and 
thumbs  on  one  glove?  How  many  on  the  two  gloves? 

How  many  nails  on  your  right  hand  ?  "  Five  nails." 
How  many  on  your  left  hand?  How  many  on  both 
hands? 

Mary  may  put  nine  shells  on  the  table,  and  Jane 
may  put  one  shell  with  Mary's.  How  many  shells  on 
the  table? 

How  many  pencils  must  I  put  with  nine  pencils  to 
make  ten  pencils? 

NOTE. — Take  different  objects  and  drill  the  pupils  until  they 
can  number  a  group  of  ten  or  less  instantly. 


2.  How  many  balls  are  nine  balls  and  one  ball? 
Eight  balls  and  two  balls?  Seven  balls  and  three 
balls?  Six  balls  and  four  balls?  Five  balls  and  five 
balls? 

NOTE.— Present  groups  of  balls  as  indicated  in  the  questions. 

How  many  pencils  are  six  pencils  and  two  pencils? 
Six  pencils  and  four  pencils? 

Charles  may  put  eight  blocks  on  the  table.  How 
many  blocks  must  I  put  with  them  to  make  ten 
blocks  ? 

How  many  blocks  are  eight  blocks  and   two  blocks? 

Jonas  may  hand  me  seven  pencils,  and  Henry  may 
hand  me  three  pencils.  How  many  pencils  have  I 
now? 


54  OEAL  LESSONS  IN  NUMBER. 

How  many  pencils  are  seven  pencils  and  three  pen- 
cils? 

Here  are  six  beautiful  roses.  How  many  roses  must 
I  put  with  them  to  make  ten  roses? 

How  many  roses  are  six  roses  and  four  roses? 

How  many  toes  on  your  right  foot  ?  How  many  toes 
on  your  left  foot?  How  many  toes  on  both  of  your 
feet? 

How  many  toes  are  five  toes  and  five  toes? 

How  many  oranges  are  eight  oranges  and  two 
oranges?  Six  oranges  and  four  oranges?  Seven 
oranges  and  three  oranges?  Five  oranges  and  five 
oranges  ? 


3.  How  many  balls  on  this  wire?  "Ten  balls." 
How  many  balls  do  I  move  away?  "Two  balls." 
How  many  balls  are  left? 

NOTE. — Continue  this  drill,  taking  from  ten  balls,  three  balls, 
four  balls,  etc. 

Willie  may  put  ten  shells  on  the  table,  and  John 
may  take  away  three  shells.  How  many  shells  are  left 
on  the  table? 

How  many  shells  are  ten  shells  less  three  shells? 

Clara  may  put  six  shells  on  the  table,  and  Martha 
may  put  four  shells  with  Clara's.  How  many  shells 
are  on  the  table? 

Martha  may  take  away  her  four  shells.  How  many 
shells  are  left  on  the  table? 

How  many  shells  are  ten  shells  less  four  shells? 

Jane  may  hand  Harry  five  pencils,  and  Kate  may 
hand  him  five.  How  many  pencils  has  Harry? 

Harry  may  give  Jane  five  of  his  pencils.  How  many 
pencils  has  Harry  left? 


FIRST- YEAR  COURSE.  55 

How  many  apples  are  ten  apples  less  one  apple? 
Ten  apples  less  two  apples?  Ten  apples  less  three 
apples?  Ten  apples  less  four  apples?  Ten  apples  less 
five  apples? 

How  many  roses  are  nine  roses  and  one  rose?  Ten 
roses  less  one  rose?  One  rose  and  nine  roses?  Ten 
roses  less  nine  roses  ? 

How  many  roses  are  eight  roses  and  two  roses?  Ten 
roses  less  two  roses  ?  Two  roses  and  eight  roses  ?  Ten 
roses  less  eight  roses? 

How  many  chickens  are  seven  chickens  and  three 
chickens?  Ten  chickens  less  three  chickens?  Three 
chickens  and  seven  chickens?  Ten  chickens  less  seven 
chickens  ? 

How  many  lambs  are  six  lambs  and  four  lambs? 
Ten  lambs  less  four  lambs?  Four  lambs  and  six 
lambs?  Ten  lambs  less  six  lambs? 

How  many  men  are  five  men  and  five  men?  Ten 
men  less  five  men  ? 


4.  Here  are  ten  red  cherries,  and  let  us  see  in  how 
many  ways  we  can  divide  them  into  two  groups  : 

(1)  Ten  cherries  are  nine  cherries  and  cherry,  or 

one  cherry  and  cherries. 

(2)  Ten  cherries  are  eight  cherries  and cherries, 

or  two  cherries  and  cherries. 

(3)  Ten  cherries  are  seven  cherries  and cherries, 

or  three  cherries  and  cherries. 

(4)  Ten  cherries  are  six  cherries  and cherries,  or 

four  cherries  and  cherries. 

(5)  Ten  cherries  are  five  cherries  and cherries. 

Here  are  ten  blocks,  ten  shells,  ten  pebbles,  ten  but- 
tons, and  ten  cherries.  Susan  may  divide  the  blocks 
into  two  groups;  Kate,  the  shells  into  two  other 


56  ORAL  LESSONS  IN  NUMBER. 

groups;  Lucy,  the  pebbles  in  two  other  groups;  Helen, 
the  buttons  into  two  other  groups;  and  Alice,  the  cher- 
ries into  two  other  groups.  Each  girl  may  now  tell  us 
the  result  of  her  division. 

James  may  take  ten  shells  and  divide  them  into  two 
groups  in  five  ways,  telling  us  after  each  division  the 
result. 

NOTE. — The  five  divisions  will  give  these  results: 

f9  shells  and  1  shell,  or  1  shell  and  9  shells. 

8  shells  and  2  shells,  or  2  shells  and  8  shells. 
Ten  shells  are  :  I  7  shells  and  3  shells,  or  3  shells  and  7  shells. 
I  6  shells  and  4  shells,  or  4  shells  and  6  shells. 
L  5  shells  and  5  shells. 

Clarence  may  take  ten  cents  and  divide  them  into 
two  groups  in  as  many  ways  as  he  can. 

Frank  may  take  ten  blocks  and  divide  them  into  two 
equal  groups.  How  many  blocks  in  each  group? 

How  many  five  blocks  make  ten  blocks? 

Lucy  may  take  ten  shells  and  divide  them  into  five 
equal  groups.  How  many  shells  in  each  group? 

How  many  two  shells  make  ten  shells? 


5.  How  many  peaches  are: 

Ten  peaches  are  nine  peaches  and peach,  or  one 

peach  and  -  -  peaches. 

Ten  peaches  less  one  peach  are peaches;  ten 

peaches  less  nine  peaches  are  peach. 

Ten  peaches  are  eight  peaches  and  -  -  peaches,  or 
two  peaches  and  peaches. 

Ten  peaches  less  two  peaches  are  -  -  peaches;  ten 
peaches  less  eight  peaches  are  -  -  peaches. 

Ten  pears  are  seven  pears  and  pears,  or  three 

pears  and  pears. 

Ten  pears  less  three  pears  are  -  -  pears;  ten  pears 
less  seven  pears  are  -  -  pears. 


FIRST- YEAR   COURSE.  57 

Ten  pears  are  six  pears  and  -  -  pears,  or  four  pears 
and  -  -  pears. 

Ten  pears  less  four  pears  are  -  -  pears;  ten  pears 
less  six  pears  are  -  -  pears. 

Ten  oranges  are  five  oranges  and  oranges;  ten 

oranges  less  five  oranges  are  oranges. 


6.  There  are  nine  birds  on  a  tree,  and  seven  birds  on 
the  ground.  How  many  birds  on  the  tree  more  than 
on  the  ground? 

I  have  ten  cents  in  my  right  hand  and  six  cents  in 
my  left  hand.  How  many  cents  in  my  right  hand 
more  than  in  my  left? 

How  many  pencils  must  be  taken  from  ten  pencils  to 
leave  seven  pencils?  To  leave  five  pencils? 

Harry  gave  ten  cents  for  a  book  and  eight  cents  for 
a  slate.  How  many  cents  did  the  book  cost  more  than 
the  slate? 

There  are  ten  sheep  in  one  field,  and  seven  sheep  in 
another  field.  How  many  sheep  in  the  first  field  more 
than  in  the  second? 

There  are  ten  girls  and  six  boys  in  a  class.  How 
many  more  girls  than  boys  in  the  class? 


7.  There  are  five  desks  in  one  row,  and  five  desks  in 
another  row.  How  many  desks  in  both  rows? 

Jane  found  six  eggs  in  one  nest  and  four  eggs  in 
another  nest.  How  many  eggs  did  she  find? 

A  boy  gave  ten  cents  for  a  slate  and  eight  cents  for  a 
book.  How  many  more  cents  did  he  pay  for  the  slate 
than  for  the  book? 


58  ORAL  LESSONS  IN  NUMBER. 

A  man  paid  three  dollars  for  a  hat  and  seven  dollars 
for  a  pair  of  boots.  How  many  dollars  did  he  pay  for 
both? 

How  many  more  dollars  did  the  boots  cost  than  the 
hat? 

Henry  earned  ten  cents,  and  paid  six  cents  for  two 
pencils.  How  many  cents  had  he  left? 

There  were  nine  cars  in  one  train,  and  seven  cars  in 
another  train.  How  many  more  cars  in  the  first  train 
than  in  the  second? 


8.  This  is  the  word  that  stands  for  ten : 
These  are  the  figures  that  stand  for  ten:  20. 
Make  10  on  your  slate  ten  times. 
Make  1,  2,  3,  4,  5,  6,  7,  8,  9,  10  on  your  slate. 

NOTE. — Teach  the  name  and  value  of  the  figure  0,  and  show 
the  position  of  the  1  and  0  in  10. 


REVIEW    EXERCISES. 

How  many  hands  are: 

Five  hands  and  two  hands? 

Seven  hands  less  two  hands? 
Six  hands  and  two  hands? 

Eight  hands  less  two  hands? 
Seven  hands  and  two  hands? 

Nine  hands  less  two  hands? 
Eight  hands  and  two  hands? 

Ten  hands  less  two  hands? 
Four  hands  and  three  hands? 

Seven  hands  less  three  hands? 


FIRST- YEAR  COURSE.  59 

Five  hands  and  three  hands? 

Eight  hands  less  three  hands? 
Six  hands  and  three  hands? 

Nine  hands  less  three  hands? 
Seven  hands  and  three  hands? 

Ten  hands  less  three  hands? 
How  many  cents  are: 

Three  cents  and  four  cents? 

Seven  cents  less  four  cents? 
Four  cents  and  four  cents? 

Eight  cents  less  four  cents? 
Five  cents  and  four  cents? 

Nine  cents  less  four  cents? 
Six  cents  and  four  cents? 

Ten  cents  less  four  cents  ? 
Two  cents  and  five  cents? 

Seven  cents  less  five  cents? 
Three  cents  and  five  cents? 

Eight  cents  less  five  cents? 
Four  cents  and  five  cents? 

Nine  cents  less  five  cents? 
Five  cents  and  five  cents? 

Ten  cents  less  five  cents? 
How  many  birds  are: 

Two  birds  and  six  birds? 

Eight  birds  less  six  birds? 
Three  birds  and  six  birds? 

Nine  birds  less  six  birds? 
Four  birds  and  six  birds? 

Ten  birds  less  six  birds? 
One  bird  and  seven  birds? 

Eight  birds  less  seven  birds? 
Two  birds  and  seven  birds? 

Nine  birds  less  seven  birds? 
Three  birds  and  seven  birds? 

Ten  birds  less  seven  birds? 


60  ORAL  LESSONS  IN  NUMBER. 


LESSON  X. 

Separate  each  number  from  1  to  10  inclusive  into  any 
two  smaller  numbers  that  compose  it,  and  then  subtract 
each  of  the  two  smaller  numbers  thus  found  from  the 
original  number,  as  follows : 

2  cents  are  1  cent  and  1  cent  /.  2  cents  less  1  cent  are 

1  cent. 

3  cents  are  2  cents  and  1  cent,  or  1  cent  and  2  cents  .'. 

3  cents  less  2  cents  are  1  cent;  3  cents  less  1  cent 
are  2  cents. 

4  cents  are  3  cents  and  1  cent,  or  1  cent  and  3  cents  .'. 

4  cents  less  3  cents  are  1  cent ;  4  cents  less  1  cent 
are  3  cents. 

4  cents  are  2  cents  and  2  cents  .'.  4  cents  less  2  cents  are 

2  cents. 

5  cents  are  4  cents  and  1  cent,  or  1  cent  and  four  cents  .'. 

5  cents  less  4  cents  are  1  cent;  5  cents  less  1  cent 
are  4  cents. 

5  cents  are  3  cents  and  2  cents,  or  2  cents  and  3  cents  /. 

5  cents  less  3  cents  are  2  cents ;  5  cents  less  2  cents 
are  3  cents. 

6  cents  are  5  cents  and  1  cent,  or  1  cent  and  5  cents  .'. 

6  cents  less  5  cents  are  1  cent ;  6  cents  less  1  cent 
are  5  cents. 

6  cents  are  4  cents  and  2  cents,  or  2  cents  and  4  cents  .*. 

6  cents  less  4  cents  are  2  cents ;  6  cents  less  2  cents 
are  4  cents. 

6  cents  are  3  cents  and  3  cents  .*.  6  cents  less  3  cents 

are  3  cents. 

7  cents  are  6  cents  and  1  cent,  or  1  cent  and  6  cents  .*. 

7  cents  less  6  cents  are  1  cent;  7  cents  less  1  cent 
are  6  cents. 


FIRST-YEAR  COURSE.  61 

7  cents  are  5  cents  and  2  cents,  or  2  cents  and  5  cents  /. 
7  cents  less  5  cents  are  2  cents;  7  cents  less  2  cents 
are  5  cents. 

7  cents  are  4  cents  and  3  cents,  or  3  cents  and  4  cents  /. 

7  cents  less  4  cents  are  3  cents ;  7  cents  less  3  cents 
are  4  cents. 

8  cents  are  7  cents  and  1  cent,  or  1  cent  and  7  cents  .'. 

8  cents  less  7  cents  are  1  cent;  8  cents  less  1  cent 
are  7  cents. 

8  cents  are  6  cents  and  2  cents,  or  2  cents  and  6  cents  .*. 

8  cents  less  6  cents  are  2  cents;  8  cents  less  2  cents 

are  6  cents. 
8  cents  are  5  cents  and  3  cents,  or  3  cents  and  5  cents  .*. 

8  cents  less  5  cents  are  3  cents ;  8  cents  less  3  cents 
are  5  cents. 

8  cents  are  4  cents  and  4  cents  /.  8  cents  less  4  cents 

are  4  cents. 

9  cents  are  8  cents  and  1  cent,  or  1  cent  and  8  cents  /. 

9  cents  less  8  cents  are  1  cent;  9  cents  less  1  cent 
are  8  cents. 

9  cents  are  7  cents  and  2  cents,  or  2  cents  and  7  cents  /. 

9  cents  less  7  cents  are  2  cents ;  9  cents  less  2  cents 

are  7  cents. 
9  cents  are  6  cents  and  3  cents,  or  3  cents  and  6  cents  .*. 

9  cents  less  6  cents  are  3  cents ;  9  cents  less  3  cents 

are  6  cents. 

9  cents  are  5  cents  and  4  cents,  or  4  cents  and  5  cents  .*. 

9  cents  less  5  cents  are  4  cents ;  9  cents  less  4  cents 
are  5  cents. 

10  cents  are  9  cents  and  1  cent,  or  1  cent  and  9  cents  /. 

10  cents  less  9  cents  are  1  cent;  10  cents  less  1  cent 
are  9  cents. 


62  ORAL   LESSONS  IN  NUMBER. 

10  cents  are  8  cents  and  2  cents,  or  2  cents  and  8  cents  .'. 

10  cents  less  8  cents  are  2  cents;    10  cents  less  2 

cents  are  8  cents. 
10  cents  are  7  cents  and  3  cents,  or  3  cents  and  7  cents  .'. 

10  cents  less   7  cents  are  3  cents;    10  cents  less  3 

cents  are  7  cents. 
10  cents  are  6  cents  and  4  cents,  or  4  cents  and  6  cents  /. 

10  cents  less  6  cents  are  4  cents;  10  cents  less  4 

cents  are  6  cents. 

10  cents  are  5  cents  and  5  cents  .'.  10  cents  less  5  cents 
are  5  cents. 

NOTE. — In  some  schools  it  may  be  deemed  best  to  give  the 
pupils  slate  exercises  the  latter  part  of  the  first  year.  Board  and 
slate  exercises  may  be  easily  provided,  and  by  mentally  adding 
the  words  balls,  blocks,  pears,  etc.,  after  the  figures,  the  numbers 
in  these  exercises  may  be  made  concrete.  The  following  board 
and  slate  exercises  will  serve  as  illustrations  : 


1.  Add 

1      7 

2 

6 

3 

5 

4 

a 

6 

c 

d 

e 

7      1 

6 

2 

5 

3 

4 

2 

2 

2 

2 

2 

2 

2 

3 

1 

1 

From 

8      8 

8 

8 

8 

8 

8 

2 

2 

2 

3 

2 

Take 

A    7 

A 

6 

c> 

_5_ 

A 

J.dc?  2 

_L 

1_ 

2 

3 

2.  Add 

1      8 

2      7 

3 

6 

4 

5 

a 

b 

c 

d 

e 

8      1 

7      2 

6 

3 

5 

4 

3 

2 

2 

3 

2 

2 

2 

3 

1 

2 

From 

9      9 

9      9 

9 

9 

9 

9 

2 

2 

2 

3 

2 

Take 

1      8 

2      7 

3 

6 

4 

5 

^eta  2 

3 

1 

2 

1 

N.  B. — When  pupils  are  clearly  able  to  advance  be- 
yond the  foregoing  lessons  the  first  year,  one  or  more 
of  the  lessons  in  the  second-year  course  may  be  taught, 
but  no  attempt  should  be  made  to  go  beyond  Lesson  V. 
The  first-year  course  should  be  thoroughly  mastered. 


SECOND-YEAR   COURSE. 


Aims. 

1.  To  teach  the  numbers  from  eleven  to  twenty  inclusive, 
and  their  representation  by  figures. 

2.  To   teach  the  adding,   subtracting,  and   analyzing   of 
numbers,  the  amounts,  minuends,  and  numbers  analyzed  not 
exceeding  twenty. 

Steps. 

The  steps  taken  to  attain  the  second  aim  are  as  fol- 
lows: 

1.  The  adding  of   any  two  digital   numbers  without 
counting,  and  the  subtracting  of  each  from  their  sum. 

2.  The    separating    of   each    number,   not    exceeding 
eighteen,  into  any  two  digital  numbers  that  compose  it, 
and  the  subtracting  of  each  number  thus  found   from 
the  original  number. 

3.  The    separating    of   any    number,    not    exceeding 
twenty,  that  is  composed  of  equal  numbers  greater  than 
1,  into  all  the  equal  numbers  that  compose  it. 

4.  The  applying  of  the  processes  learned  to  the  solu- 
tion of  practical  problems,  involving  a  simple  exercise 
of  imagination  and  judgment. 

5.  Blackboard   and   slate   exercises    in    addition    and 
subtraction,    amounts    and    minuends    not     exceeding 
twenty. 

(63) 


04  ORAL  LESSONS  IN  NUMBER. 

LESSON  I. 

The  Numbers  11  to  15. 

NOTE.— The  object  of  this  lesson  is  to  develop  the  idea  of  each 
number  from  eleven  to  fifteen  inclusive,  to  teach  its  name,  and  its 
representation  by  figures.  The  teacher  should  be  supplied  with 
a  numeral  frame  (or,  in  its  absence,  with  twenty  balls  on  two 
wires,  ten  on  each),  ten  pencils  or  rods  tied  together  in  a  bunch, 
and  ten  loose  pencils  or  rods,  and  also  a  blackboard  or  large  slate. 
The  objects  referred  to  in  each  question  should  be  presented,  and 
the  action  should  otherwise  be  suited  to  the  word. 

1,  How  many  balls  on  this  wire?  "Ten  balls." 
How  many  balls  do  I  now  move  below  the  ten  balls? 
"  One  ball."  Ten  balls  and  one  ball  are  eleven  balls. 

If  I  take  one  ball  from  the  eleven  balls,  how  many 
balls  will  be  left? 

How  many  pencils  are  ten  pencils  and  one  pencil? 
Eleven  pencils  less  one  pencil? 

How  many  blocks  are  ten  blocks  and  one  block? 
Eleven  blocks  less  one  block? 

I  will  make  ten  lines  on  the  board.  How  many  lines 
more  will  make  eleven  lines? 

NOTE. — Make  ten  lines  on  the  board,  and  then  add  one  line  at 
the  right  or  below. 

If  I  rub  out  one  line,  how  many  lines  will  be  left? 

Here  are  the  figures  that  stand  for  eleven  :  21. 

Make  eleven  lines  on  your  slate,  and 
write  below  them  the  figures  that  stand 
for  eleven.  11 


2.  Here  are  ten  balls,  and  I  now  move  two  balls 
beneath  them.  Ten  balls  and  two  balls  are  twelve  balls. 

If  I  take  two  balls  from  twelve  balls,  how  many  balls 
will  be  left? 


SECOND-YEAR   COURSE.  65 

How  many  pencils  are  ten  pencils  and  two  pencils? 
Twelve  pencils  less  two  pencils? 

How  many  blocks  are  ten   blocks   and  two  blocks? 
Ten  lines  and  two  lines? 

How  many  are  twelve  blocks  less  two  blocks  ?    Twelve 
lines  less  two  lines? 

Here  are  the  figures  that  stand  for  twelve:  2@. 

Make  twelve  lines  on  your  slate,  and 
write  the  figures  that  stand  for  twelve 
below  them.  12 

NOTE. — Show  that  the  figure  1  stands  for  one  ten  (ten  balls, 
ten  pencils,  etc.),  and  the  figure  2  for  the  two  added. 


3.  I  now  move  three  balls  beneath  the  ten  balls. 
Ten  balls  and  three  balls  are  thirteen  balls. 

How  many  balls  are  ten  balls  and  one  ball?  Ten 
balls  and  two  balls?  Ten  balls  and  three  balls? 

How  many  pencils  are  ten  pencils  and  three  pencils? 
Thirteen  pencils  less  three  pencils? 

How  many  balls  are  ten  balls  and  three  balls?  Ten 
blocks  and  three  blocks?  Ten  lines  and  three  lines? 

How  many  are  thirteen  balls  less  three  balls?  Thir- 
teen blocks  less  three  blocks?  Thirteen  lines  less  three 
lines. 

Here  are  the  figures  that  stand  for  thirteen :  23. 

Make  thirteen  lines  on  your  slate,  and 
write  below  them  the  figures  that  stand 
for  thirteen.  13 

NOTE. — Show  that  the  figure  1  stands  for  ten,  and  the  figure  3 
for  three  ones. 

Make  the  figures  11,  12,  and  13  on  your  slate. 

O.  L. — 6. 


66  ORAL  LESSONS  IN  NUMBER. 

4.  I  now  move  four  balls  below  the  ten  balls.  Ten 
balls  and  four  balls  are  fourteen  balls. 

If  four  balls  are  taken  from  fourteen  balls,  how  many 
balls  will  be  left? 

How  many  pencils  are  ten  pencils  and  four  pencils? 
Fourteen  pencils  less  four  pencils? 

How  many  are  ten  blocks  and  four  blocks?  Ten 
shells  and  four  shells?  Ten  lines  and  four  lines? 

How  many  are  fourteen  blocks  less  four  blocks? 
Fourteen  shells  less  four  shells?  Fourteen  lines  less 
four  lines? 

Here  are  the  figures  that  stand  for  fourteen  :  1^,, 

NOTE. — Show  that  the  1  stands  for  one  ten,  and  the  4  for  four 
ones.  Illustrate  by  one  dime  and  four  cents. 


Make    fourteen    lines    on    your   slate, 
and  write  14  below  them.  14 

Write  11,  12,  13,  14  on  your  slate. 


5.  I  now  move  five  balls  beneath  the  ten  balls.  Ten 
balls  and  five  balls  are  fifteen  balls. 

Five  balls  taken  from  fifteen  balls  leave1  how  many 
balls? 

How  many  pencils  are  ten  pencils  and  five  pencils? 
Fifteen  pencils  less  five  pencils? 

How  many  are  ten  shells  and  five  shells?  Ten  lines 
and  five  lines?  Ten  cents  and  five  cents? 

How  many  are  fifteen  shells  less  five  shells?  Fifteen 
lines  less  five  lines?  Fifteen  cents  less  five  cents? 

Here  are  the  figures  that  stand  for  fifteen :  15. 

NOTE. — Show  that  the  figure  1  stands  for  one  ten,  and  the 
figure  5  for  five  ones.  This  may  be  illustrated  by  taking  fifteen 
cents, — one  dime  and  five  cents.  The  dime  or  ten-cent  piece 


SECOND-YEAR  COURSE.  67 

equals  in  value  ten  cents,  and  represents  one  ten  or  ten  ones, 
and  the  five  cents  represent  five  ones.  The  dime  and  five  cents 
represent  fifteen  ones. 


Make  fifteen  lines  on  your  slate,  and 
write  15  below  them.  15 

Make  11,  12,  13,  14,  15  on  your  slate. 

NOTE.— If  the  pupils  are  specially  bright,  and  are  interested, 
the  teacher  can  profitably  continue  these  exercises,  teaching.the 
numbers  sixteen,  seventeen,  etc.,  to  twenty ;  but,  inasmuch  as 
no  number  exceeding  fifteen  will  be  used  for  several  weeks,  it  is 
believed  to  be  wise  to  defer  the  teaching  of  these  higher  numbers 
until  the  sum  of  the  two  digital  numbers  used  in  any  exercise 
is  fifteen  or  more.  See  Lesson  VII,  page  88. 


LESSON  II. 

1.  John  may  take  2  books  in  his  right  hand,  and  1 
book  in  his  left  hand.  How  many  books  has  John  ? 

How  many  books  are  1  book  and  1  book?  2  books 
and  1  book?  3  books  and  1  book? 

How  many  are  1  and  1?    2  and  1?     3  and  1? 

Kate  had  3  plums,  and  she  gave  1  plum  to  her 
brother.  How  many  plums  had  Kate  left? 

How  many  plums  are  2  plums  less  1  plum?  3 
plums  less  1  plum?  4  plums  less  1  plum? 

How  many  are  2  less  1  ?    3  less  1  ?    4  less  1  ?    5  less  1  ? 


2.  Charles  has  4  marbles  in  his  right  hand  and  1  in 
his  left.  How  many  marbles  has  Charles  in  both 
hands  ? 

How  many  marbles  are  4  marbles  and  1  marble?  5 
marbles  and  1  marble?  6  marbles  and  1  marble? 

How  many  are  4  and  1  ?    5  and  1  ?     6  and  1  ? 


68 


ORAL  LESSONS  IN  NUMBER. 


John  has  6  marbles:  if  he  should  give  1  marble  to 
Willie,  how  many  marbles  would  he  have  left? 

How  many  marbles  are  5  marbles  less  1  marble?  6 
marbles  less  1  marble?  7  marbles  less  1  marble? 

How  many  are  5  less  1?    6  less  1?    7  less  1? 


2.  Kate  gave  7  cents  for  paper  and  1  cent  for  a  pencil. 
How  many  cents  did  she  give  for  paper  and  pencil? 

How  many  cents  are  7  cents  and  1  cent?  8  cents 
and  1  cent?  9  cents  and  1  cent? 

How  many  are  7  and  1?     8  and  1?    9  and  1? 

Mary  had  10  cents,  and  she  gave  1  cent  for  a  pencil. 
I  low  many  cents  had  she  left? 

How  many  cents  are  8  cents  less  1  cent?  9  cents 
less  1  cent?  10  cents  less  1  cent? 

How  many  are  8  less  1  ?    9  less  1  ?     10  less  1  ? 

How  many  are  4  cents  and  1  cent?  5  cents  less  1 
cent?  7  cents  and  1  cent?  8  cents  less  1  cent?  9 
cents  and  1  cent?  10  cents  less  1  cent? 

How  many  are  5  and  -1  ?  6  less  1  ?  8  and  1  ?  9  less 
1  ?  4  and  1  ?  5  less  1  ?  9  and  1  ?  10  less  1  ?  7  and 
1  ?  8  less  1  ?  6  and  1  ?  7  less  1  ? 


4.  How 

many  are: 

Read  and  complete: 

1 

and 

1 

? 

2 

less 

1? 

1 

-fl  =  2 

2  — 

1=1 

2 

and 

1 

? 

3 

less 

1? 

2 

-f  1  = 

3- 

1  = 

3 

and 

1 

? 

4 

less 

1? 

3 

+  1  = 

4- 

1  = 

4 

and 

1 

? 

5 

less 

1? 

4 

+  1  — 

5- 

1  = 

5 

and 

1 

? 

6 

less 

1? 

5 

i   -«  . 

6- 

1  = 

6 

and 

1 

? 

7 

less 

1? 

6 

4-  1  = 

7- 

1  — 

7 

and 

1 

? 

8 

less 

1? 

7 

_l_  i  — 

8  — 

1  = 

8 

and 

1 

? 

9 

less 

1? 

8 

+•1  = 

9- 

1  = 

9 

and 

1 

? 

10 

less 

1? 

9 

4-1  = 

10  — 

-f 

SECOND-YEAR  COURSE.  69 

NOTES. — 1.  The  left-hand  tables  may  first  be  recited  together, 
thus:  1  and  1  are  2;  2  less  1  is  1,  etc.  They  should  then  be 
recited  separately,  and  the  drill  should  be  continued  until  they 
are  each  recited  without  hesitation. 

2.  Copy  the  two  right-hand  tables  on  the  blackboard,  and 
after  teaching  the  signs  +,  —  and  — ,  drill  until  the  pupils  can 
read  and  complete  in  order  or  promiscuously  with  rapidity. 
The  sign  -f  may  be  read  plus  or  and,  as  may  be  preferred,  and 
the  sign  — ,  less.  The  two  tables  should  finally  be  copied  by  the 
pupils  on  their  slates  and  completed. 


LESSON   III. 

1.  I  have  2  books  in  my  right  hand,  and  2  books  in 
my  left.  How  many  books  in  both  of  my  hands? 

How  many  books  are  1  book  and  2  books?  2  books 
and  2  books?  3  books  and  2  books? 

How  many  are  1  and  2?    2  and  2?     3  and  2? 

John  picked  3  pears,  and  gave  2  of  them  to  his 
mother.  How  many  pears  had  John  left? 

How  many  pears  are  3  pears  less  2  pears?  4  pears 
less  2  pears?  5  pears  less  2  pears? 

How  many  is  2  less  2  ?    3  less  2  ?    4  less  2  ?    5  less  2  ? 


.  James  found  4  eggs  in  one  nest  and  2  eggs  in  another 
nest.  How  many  eggs  did  James  find? 

How  many  eggs  are  4  eggs  and  2  eggs?  5  eggs  and  2 
eggs?  6  eggs  and  2  eggs? 

How  many  are  4  and  2?    5  and  2?     6  and  2? 

Jane  picked  7  plums,  and  gave  2  of  them  to  Susan. 
How  many  plums  had  Jane  left? 

How  many  plums  are  6  plums  less  2  plums?  7 
plums  less  2  plums?  8  plums  less  2  plums? 

How  many  are  6  less  2  ?    7  less  2  ?    8  less  2  ? 


70 


ORAL  LESSONS  IN  NUMBER. 


2,  There  are  7  birds  on  a  branch  of  a  tree,  and  2 
birds  on  another  branch.  How  many  birds  on  the 
two  branches? 

How  many  birds  are  7  birds  and  2  birds?  8  birds 
and  2  birds?  9  birds  and  2  birds? 

How  many  are  7  and  2?    8  and  2?    9  and  2? 

There  are  10  birds  on  a  tree.  If  2  birds  fly  away, 
how  many  birds  will  be  left  ? 

How  many  birds  are  10  birds  less  2  birds?  9  birds 
less  2  birds?  11  birds  less  2  birds? 

How  many  are  9  less  2?     10  less  2?     11  less  2? 

How  many  are  5  cents  and  2  cents?  7  cents  less  2 
cents?  7  cents  and  2  cents?  9  cents  less  2  cents? 

How  many  are  6  cents  and  2  cents?  8  cents  less  2 
cents?  4  cents  and  2  cents?  6  cents  less  2  cents? 


3.  How  many  are: 

Read  and  complete  : 

1  and  2  ? 

3  less  2  ? 

1  +  2  = 

3 

—  2 

2  and  2  ? 

4  less  2  ? 

2  +  2  = 

4 

2 

3  and  2? 

5  less  2  ? 

3  +  2  = 

5 

2 

4  and  2? 

6  less  2  ? 

4  +  2  = 

6 

—  2 

5  and  2  ? 

7  less  2? 

5  +  2  = 

7 

—  2 

6  and  2  ? 

8  less  2? 

6  +  2  = 

8 

—  2 

7  and  2? 

9  less  2  ? 

7  +  2  = 

9 

—  2 

8  and  2? 

10  less  2? 

8  +  2  = 

10 

2 

9  and  2  ? 

11  less  2? 

9  +  2  = 

11 

—  2 

NOTE. — For  directions,  see  page  69,   notes  1  and  2. 

7  and  2? 


How  many  are  3  and  2?  5  and  2? 
and  2?  4  and  2?  6  and  2?  9  and  2? 

How  many  are  9  and  2?  11  less  2? 
less  2?  5  and  2?  7  less  2?  8  and  2? 


8 


7  and  2?     9 
10  less  2? 


How  many  are  8  and  2?     10  less  2?    7  less  2?    6 
less  2?    9  less  2?    5  less  2?     11  less  2? 


SECOND-YEAR  COURSE.  71 

4.  Separate  each  of  the  numbers  from  3  to  11  in- 
clusive into  2  and  another  number,  and  then  subtract 
2  and  its  complement  from  the  original  number,  thus : 

3  is  2  and — ,  or  —  and  2  .*.  3  less  2  is  — ;  3  less  —  is  2. 

4  is  2  and  —  .'.  4  less  2  is  — . 

5  is  2  and  — ,  or  —  and  2  .*.  5  less  2  is  — ;  5  less  —  is  2. 

6  is  2  and  — ,  or  —  and  2  /.  6  less  2  is  —  ;  6  less  —  is  2. 

7  is  2  and  — ,  or  —  and  2  /.  7  less  2  is  — ;  7  less  —  is  2. 

8  is  2  and  — ,  or  —  and  2  /.  8  less  2  is  — :  8  less  —  is  2. 

9  is  2  and — ,  or  —  and  2 /.  9  less  2  is  — ;  9  less  —  is  2. 

10  is  2  and  — ,  or  —  and  2  .-.  10  less  2  is  — ;  10  less  —  is  2. 

11  is  2  and  — ,  or  —  and  2  .\  11  less  2  is  — ;  11  less  —  is  2. 

NOTES. — 1.  The  character  .*.  used  in  the  above  exercises  is 
read  whence  or  hence.  It  may  be  omitted  in  reciting  these  exer- 
cises. 

2.  These  two  series  of  inverse  exercises  may  also  be  written 
on  the  blackboard  and  combined  in  one  drill  (the  pupils  filling 
the  blanks),  thus: 

3  =  2+  ,  or  +2  .'.  3  —  2=  ;  3-  =2. 

4  =  2+  .*.  4  —  2=  . 

5  =  2+  ,  or  +2  .'.  5  —  2=  ;  5—  =2. 

6  =  2+  ,  or  +2  /.  6  —  2=  ;  6-  =2. 

7  =  2+  ,  or  +2  .-.  7  —  2=  ;  7—  =2. 

8  =  2+  ,  or  +2  /.  8  —  2=  ;  8—  =2. 

9  =  2+  ,  or  +2  /.  9  —  2=  ;  9—  =2. 

10  =  2+     ,   or        +2     .\     10  —  2=     ;     10—       =2. 

11  =  2+     ,   or        +2     .-.     11—2=     ;     11—       =2. 


5.  How  many  books  are  2  books  +  2  books  +  2 
books?  How  many  books  are  three  2  books?  "  Six 
books." 

How  many  are  two  2's?  Three  2's?  Four  2's?  Five 
2's? 


72  ORAL  LESSONS  IN  NUMBER. 

How  many  2  books  make  4  books?  6  books?  8 
books?  10  books?  12  books? 

NOTE. — Take  4  books,  6  books,  etc.,  and  separate  them  into 
groups  of  2  books. 

How  many  2's  in  4?    2's  in  6?    2's  in  8?    2's  in  10? 

2's  in  12? 

NOTE.— Give  the  above  and  subsequent  similar  exercises  with- 
out introducing  the  word  "  times "  or  the  idea  of  factor.  The 
equal  numbers  should  be  added  or  combined  as  parts,  and  the 
numbers  should  be  separated  into  equal  parts.  These  exercises 
not  only  afford  practice  in  addition  and  analysis  of  numbers, 
but  they  will  prepare  the  way  for  multiplication  and  division. 
It  should  be  kept  in  mind  that  the  adding  of  the  equal  parts 
of  a  number  is  not  multiplication,  and  the  separating  of  a  num- 
ber into  its  equal  parts  is  not  numerical  division.  See  page  15. 


6.  Jane  picked  5  red  roses  from  one  bush,  and  2 
white  roses  from  another  bush.  How  many  roses  did 
she  pick? 

Willie  earned  10  cents  by  selling  papers,  and  then 
gave  2  cents  for  a  pencil.  How  many  cents  had  he 
left? 

NOTE. — Do  not  teach  any  formal  analysis  of  these  and  subse- 
quent similar  problems  in  this  second-year  course,  and  do  not  re- 
quire pupils  to  use  "because,"  "since,"  "therefore,"  etc.  All 
that  should  be  given  is  the  answer  and  the  process  which  gives 
it,  as  follows: 

(1)  Seven  roses:  5  roses  and  2  roses  are  7  roses. 

(2)  Eight  cents:  10  cents  less  2  cents  are  8  cents. 

A  train  of  cars  contains  7  passenger  cars  and  2 
baggage  cars.  How  many  cars  in  the  train? 

A  house  has  5  rooms  on  the  first  floor  and  2  rooms 
on  the  second  floor.  How  many  rooms  in  the  house? 


SECOND-YEAR  COURSE.  73 

Mary  is  6  years  old,  and  her  brother  is  2  years  older 
than  she.  How  old  is  her  brother? 

Carrie  wrote  9  words  on  her  slate,  and  then  rubbed 
out  2  of  them.  How  many  words  were  left  on  her 
slate  ? 

A  man  brought  10  melons  to  market,  but  2  of  them 
were  stolen  by  a  wicked  boy.  How  many  melons  had 
he  left? 

A  farmer  having  11  hogs,  sold  2  of  them.  How  many 
hogs  had  he  left? 

Charles  has  9  marbles  and  John  7  marbles.  How 
many  marbles  has  Charles  more  than  John  ? 

How  many  times  2  bananas  make  8  bananas? 

BOARD   AND   SLATE   EXERCISES. 

a    b    c    d    e    f     g  a    b     c    d    e    f 

2222222  221222 

Add  _3  J[  j4    6    7    9    8  222122 

221211 
222112 

a    b    c     d    e     f      g  221222 

From    5    6    7    8    9    11     10      Add  _2_  J_  2_  2^  _2  J: 
Take     22222      2      2 


NOTES. — 1.  Write  these  exercises  on  the  blackboard,  and  drill 
the  pupils  until  the  results  are  given  instantly.  The  two  left- 
hand  series  of  exercises  will  be  sufficient  for  one  drill,  and  the 
right-hand  or  "  column  "  series  will  afford  another  drill. 

2.  In  adding  the  numbers  in  the  column  series,  give  only  re- 
sults, thus  (a) :  4,  6,  8,  10,  12.    The  pupils  should  not  be  per- 
mitted to  say  (a)  2  and  2  are  4 ;  4  and  2  are  6 ;  6  and  2  are  8, 
etc. 

3.  Have  the  pupils  copy  the  above  exercises  on  their  slates, 
add  or  subtract  the  numbers,  as  the  case  may  be,  and  write  the 

sum  or  difference  below. 
O.  L.— 7. 


74 


ORAL  LESSONS  IN  NUMBER. 


The  following  WHEEL  EXERCISES  may  be  copied  on 
the  blackboard  and  used  to  supplement  the  above 
drills : 

?  f 


Q     __ 

2                                       A 

\ 

\ 

1fi                          « 

/ 

0 

/ 
/ 
/ 

^ 

~  4: 

IS 

\/ 
1 

V 

N14 

2 

NOTES. — 1.  Point  successively  to  the  outer  numbers  in  the 
left-hand  diagram,  and  have  the  pupils  add  to  the  numbers 
thus  designated  the  number  2  in  the  center. 

2.  In  the  use  of  the  right-hand  diagram,  the  number  2  in 
the  center  is  subtracted  from  the  outer  numbers  successively 
designated. 


LESSON  IV. 

1.  Mary  picked  2  pears  from  one  limb  and  3  pears 
from  another  limb.  How  many  pears  did  she  pick? 

How  many  pears  are  2  pears  and  3  pears?  1  pear 
and  3  pears?  3  pears  and  3  pears? 

How  many  are  1  and  3?     2  and  3?    3  and  3? 

Mary  picked  5  pears,  and  gave  3  of  them  to  her 
father.  How  many  pears  had  she  left? 

How  many  pears  are  5  pears  less  3  pears?  4  pears 
less  3  pears?  6  pears  less  3  pears? 

How  many  are  4  less  3?    5  less  3?    6  less  3? 


2.  Charles  found  4  eggs  in  one  nest  and  3  eggs  in 
another  nest.  How  many  eggs  did  he  find? 

How  many  eggs  are  4  eggs  and  3  eggs?  5  eggs  and  3 
eggs?  6  eggs  and  3  eggs? 


SECOND-YEAR   COURSE. 


75 


How  many  are  4  and  3?    5  and  3?     6  and  3? 

Willie  bought  7  peaches,  and  gave  3  of  them  to  his 
playmate.  How  many  peaches  had  he  left? 

How  many  peaches  are  7  peaches  less  3  peaches?  8 
peaches  less  3  peaches?  9  peaches  less  3  peaches? 

How  many  are  7  less  3  ?    8  less  3  ?     9  less  3  ? 


3.  7  birds  light  on  a  bush,  and  3  birds  light  on 
another  bush.  How  many  birds  light  on  both  bushes? 

How  many  birds  are  7  birds  and  3  birds?  8  birds 
and  3  birds?  9  birds  and  3  birds? 

How  many  are  7  and  3?     8  and  3?    9  and  3? 

George  earned  10  cents,  and  gave  3  cents  for  a  lead 
pencil.  How  many  cents  had  he  left? 

How  many  cents  are  10  cents  less  3  cents?  11  cents 
less  3  cents?  12  cents  less  3  cents? 

How  many  are  10  less  3?     11  less  3?     12  less  3? 

How  many  pens  are  5  pens  and  3  pens  ?  7  pens  and 
3  pens?  8  pens  and  3  pens?  9  pens  and  3  pens? 

How  many  pens  are  6  pens  less  3  pens  ?  8  pens  less 
3  pens?  5  pens  less  3  pens?  7  pens  less  3  pens?  9 
pens  less  3  pens?  10  pens  less  3  pens? 


4.  How  many  are  : 

Read  and  complete  : 

1  and  3? 

4  less  3? 

1  +  3  = 

4  —  3 

2  and  3  ? 

5  less  3  ? 

O      1      O  

5  —  3 

3  and  3  ? 

6  less  3  ? 

3|       0 
~T~  "  

6  —  3 

4  and  3  ? 

7  less  3? 

4  +  3  = 

7  —  3 

5  and  3? 

8  less  3? 

5  +  3  = 

8  —  3 

6  and  3  ? 

9  less  3  ? 

6  +  3  = 

9  —  3 

7  and  3? 

10  less  3? 

7  +  3  = 

10  —  3 

8  and  3  ? 

11  less  3  ? 

8  +  3  = 

11  —  8 

9  and  3  ? 

12  less  3  ? 

Q      1      O  

12  —  3 

76  ORAL  LESSORS  IX  DUMBER 

How  many  are  4  and  3?  6  and  3?  8  and  3?  5 
and  3?  7  and  3?  9  and  3?  2  and  3?  3  and  3? 

How  many  are  6  and  3?  9  less  3?  8  and  3?  11 
less  3?  7  and  3?  10  less  3?  9  and  3?  12  less  3? 

How  many  are  5  and  3?  8  less  3?  10  less  3?  12 
less  3?  9  less  3?  7  less  3?  4  less  3?  6  less  3? 


5.  Separate  each  of  the  numbers,  from  4  to  12  inclu- 
sive, into  3  and  another  number,  and  then  subtract  3 
and  its  complement  from  the  original  number,  thus : 

4  is  3  and — ,  or  —  and  3  /.  4  less  3  is  — ;  4  less  —  is  3. 

5  is  3  and  — ,  or  —  and  3  /.  5  less  3  is  — ;  5  less  —  is  B. 

6  is  3  and—  /.  61es&3is— . 

7  is  3  and  — ,  or  —  and  3  /.  7  less  3  is  — ;  7  less  —  is  3. 

8  is  3  and — ,  or  —  and  3  /.  8  less  3  is — ;  8  less  —  is  3. 

9  is  3  and  — ,  or  —  and  3  /.  9  less  3  is  — ;  9  less  —  is  3. 

10  is  3  and— ,  or  —  and  3  /.  10  less  3  is— ;  10  less  —  i> :'. 

11  is  3  and  — ,  or  —  and  3  /.  11  less  3  is  — ;  11  less  —  is  3. 

12  is  3  and  — ,or  — and3  /.  121ess3is  — ;  12  less  — is 3. 

NOTE. — See  Lesson  III,  page  71,  notes,  for  directions  respect- 
ing the  use  of  the  above  exercises  as  blackboard  drills. 


6.  How  many  blocks  are  3  blocks  and  3  blocks?  3 
blocks,  3  blocks,  and  3  blocks? 

How  many  blocks  are  three  3  blocks?  "Nine 
blocks." 

How  many  are  two  3's?     Three  3's?    Four  3's? 

How  many  3  blocks  make  6.  blocks?  9  blocks?  12 
blocks? 

NOTE.— Take  6  blocks,  9  blocks,  and  12  blocks,  and  divide 
them  into  groups  of  3  blocks. 


SECOND-YEAR  COURSE,  77 

How  many  3's  in  6?    3's  in  9?    3's  in  12? 
How   many  3  eggs  in  7  eggs,  and   how  many   eggs 
over? 

How  many  3  eggs  in  10  eggs,  and  how  many  over? 
How  many  3  eggs  in  8  eggs,  and  how  many  over? 
How  many  3  eggs  in  11  eggs,  and  how  many  over  ? 

NOTE.— See  Lesson  III,  page  72,  note. 


7.  There  are  6  birds  on  the  ground  and  3  birds  on 
the  tree.  How  many  birds  in  both  places? 

ANS,  :  Nine  birds :  6  birds  and  3  birds  are  9  birds. 

There  are  8  boys  and  3  girls  in  a  class.  How  many 
pupils  in  the  class? 

A  house  has  8  windows  in  the  first  story  and  3  win- 
dows in  the  second  story.  How  many  windows  in  the 
house  ? 

A  farmer  brought  7  sacks  of  wheat  and  3  sacks  of 
corn  to  a  mill.  How  many  sacks  of  grain  did  he 
bring?  How  many  more  sacks  of  wheat  than  of  corn? 

Susan  wrote  12  words  on  her  slate,  but  misspelled  3 
words.  How  many  words  did  she  write  correctly? 

Horace  earned  10  cents,  and  then  gave  3  cents  for  a 
top.  How  many  cents  had  he  left? 

Clarence  is  11  years  old,  and  Henry  8  years  old. 
How  much  older  is  Clarence  than  Henry? 

Willie  had  12  marbles,  and  gave  3  of  them  to  James. 
How  many  marbles  had  he  left? 

Willie  then  gave  3  marbles  to  Harry.  How  many 
marbles  had  he  left? 

He  next  gave  3  marbles  to  Clarence.  How  many 
marbles  had  he  then  left? 

A  man  paid  13  dollars  for  an  overcoat,  and  3  dollars 
for  a  hat.  How  many  dollars  did  he  pay  for  the  coat 
more  than  for  the  hat? 


78 


ORAL  LESSONS  IN  NUMBER. 


SLATE    AND    BOARD    EXERCISES. 


abcdefghi 

333333333 

Add  241357689 


abcdefgh  i 
From  5  7  4  6  8  10  9  11  12 
Take  33333  33  3  3 


12 


d    e 
32 


32223 
33332 
33323 
23232 
Add  11121 


NOTE. — See  Lesson  III,  page  73,  notes. 

2  5 


3, 

/5 

6, 

-8 

X 

X 

X 

/ 

X 

f 

X 

/ 

X 

f 

X 

/ 

X 

f 

X 

/ 

\ 

s 

g  ; 

j  4 

11  ^ 

\  7 

s 

» 

J 

X 

f 

x 

/ 

X 

s 

/. 

x 

/ 

\ 

?' 

N7 

I/ 

\0 

6  9 

NOTE. — See  Lesson  III,  page  74,  notes. 


LESSON  V. 

1.  There  are  two  boys  sitting  on  one  seat,  and  4  boys 
on  another  seat.  How  many  boys  on  both  seats? 

How  many  boys  are  1  boy  and  4,  boys  ?  2  boys  and 
4  boys?  3  boys  and  4  boys? 

How  many  are  1  and  4?    2  and  4?    3  and  4? 

There  are  6  boys  sitting  on  a  seat.  If  4  of  the  boys 
go  away,  how  many  boys  will  be  left? 

How  many  boys  are  5  boys  less  4  boys  ?  6  boys  less 
4  boys?  7  boys  less  4  boys? 

How  many  are  5  less  4?    6  less  4?    7  less  4? 


SECOND-YEAR  COURSE.  79 

2.  How  many  shells  are  4  shells  and  4  shells?  8 
shells  less  4  shells?  5  shells  and  4  shells?  9  shells  less 
4  shells  ?  6  shells  and  4  shells  ?  10  shells  less  4  shells  ? 

NOTE. — If  the  pupils  hesitate  in  giving  these  results,  use  the 
shells,  combining  the  groups  and  taking  away  a  group  of  four, 
as  indicated. 

How  many  quails  are  5  quails  and  4  quails?  4 
quails  and  4  quails?  6  quails  and  4  quails? 

How  many  are  4  and  4?    5  and  4?     6  and  4? 

A  hunter  shot  5  quails  in  one  field  and  4  quails  in 
another  field.  How  many  quails  did  he  shoot  in  both 
fields? 

A  hunter  had  8  quails,  and  he  gave  4  of  them  to 
a  neighbor.  How  many  quails  had  he  left? 

How  many  quails  are  8  quails  less  4  quails?  9 
quails  less  4  quails?  10  quails  less  4  quails? 

How  many  are  8  less  4?    9  less  4?     10  less  4? 


3.  How  many  shells  are  7  shells  and  4  shells?  11 
shells  less  4  shells?  8  shells  and  4  shells?  12  shells 
less  4  shells?  9  shells  and  4  shells?  13  shells  less  4 
shells  ? 

NOTE. — Do  not  permit  the  pupils  to  count  the  shells  in  the 
groups  formed  by  combining  8  shells  and  4  shells,  or  9  shells 
and  4  shells.  If  they  can  not  number  the  group  at  sight,  show 
them  how  "to  pick  out"  ten  shells,  and  then  combine  the  10 
shells  with  3  shells  or  4  shells,  as  the  case  may  be.  A  little 
practice  will  enable  children  thus  to  number  a  group  of  11  to 
18  objects  at  sight. 

A  farmer  has  7  cows  in  one  field  and  4  cows  in 
another  field.  How  many  cows  in  both  fields? 

How  many  cows  are  7  cows  and  4  cows  ?  8  cows  and 
4  cows?  9  cows  and  4  cows? 


80 


ORAL  LESSONS  IN  NUMBER. 


How  many  are  7  and  4?     8  and  4?     9  and  4? 

A  farmer  has  11  cows  in  two  fields.  If  there  are  4 
cows  in  one  of  the  fields,  how  many  cows  are  in  the 
other  field? 

How  many  cows  are  11  cows  less  4  cows?  12  cows 
less  4  cows?  13  cows  less  4  cows? 

How  many  are  11  less  4?     12  less  4?     13  less  4? 

How  many  cents  are  5  cents  and  4  cents?  9  cents 
less  4  cents?  7  cents  and  4  cents?  11  cents  less  4 
cents?  9  cents  and  4  cents?  13  cents  less  4  cents? 

How  many  cents  are  4  cents  and  4  cents?  8  cents 
less  4  cents?  6  cents  and  4  cents?  10  cents  less  4 
cents?  8  cents  and  4  cents?  12  cents  less  4  cents? 


4.  How  many  are: 

Read  and  complete: 

1  and  4?          5  less  4? 

1+4=          5—4 

2  and  4?          6  less  4? 

2  +  4=-          6  —  4 

3  and  4?          7  less  4? 

3  +  4=          7  —  4 

4  and  4?          8  less  4? 

4  +  4=          8  —  4 

5  and  4?          9  less  4? 

5  +  4=          9  —  4 

6  and  4?        10  less  4? 

6  +  4=        10  —  4 

7  and  4?        11  less  4? 

7  +  4=        11—4 

8  and  4?        12  less  4? 

8  +  4=        12  —  4 

9  and  4?        13  less  4? 

9  +  4=        13  —  4 

How  many  are  7  and  4?     11  less  4?    6  and  4?     10 
less  4?    8  and  4?     12  less  4?    9  and  4?    13  less  4? 


How  many  are  3  and  4? 
and  4?  2  and  4?  4  and  4? 

How  many  are  6  and  4? 
less  4?  12  less  4?  7  less  4? 
less  4? 


5  and  4?    7  and  4?    9 

6  and  4?    8  and  4? 

10  less  4?    8  less  4?    6 
9  less  4?     11  less  4?     13 


SECOND-YEAR   COURSE.  81 

5.  Separate  each  of  the  numbers,  from  4  to  13  inclu- 
sive, into  4  and  another  number,  and  then  subtract  4 
and  its  complement  from  the  original  number,  thus  : 

5  is  4  and  — ,  or  —  and  4  .*.  5  less  4  is  — ;  5  less  —  is  4. 

6  is  4  and  — ,  or  —  and  4  /.  6  less  4  is  — ;  6  less  —  is  4. 

7  is  4  and  — ,  or  —  and  4  .*.  7  less  4  is  — ;  7  less  —  is  4. 

8  is  4  and  —  /.  8  less  4  is  — . 

9  is  4  and  — ,  or  —  and  4  .*.  9  less  4  is  —  ;  9  less  —  is  4. 

10  is  4  and  — ,  or  —  and  4  /.  10  less  4  is  — ;  10  less  —  is  4. 

11  is  4  and  — ,  or  —  and  4  /.  11  less  4  is  — ;  11  less  —  is  4. 

12  is  4  and  — ,  or  —  and  4  /.  12  less  4  is  — ;  12  less  —  is  4. 

13  is  4  and  — ,  or  —  and  4  /.  13  less  4  is  — ;  13  less  —  is  4. 

NOTE.— See  Lesson  III,  page  7.1,  notes,  for  directions  for  black- 
board drills. 


6.  How  many  fingers  are  4  fingers  and  4  fingers?  4 
fingers  and  4  fingers  and  4  fingers? 

How  many  are  two  4  fingers?     Three  4  fingers? 

How  many  are  two  4's?     Three  4's? 

How  many  4  shells  in  8  shells?  How  many  4  shells 
in  12  shells? 

NOTE.— Take  8  shells  and  12  shells,  and  divide  them  into 
groups  of  4  shells. 

How  many  4's  in  8?     4's  in  12? 
How  many  4  pins  in  6  pins,  and  how  many  over? 
How  many  4  pins  in  9   pins,  and  how  many  over? 
How  many  4  pins  in  10  pins,  and  how  many  over? 
How  many  4  pins  in  14  pins,  and  how  many  over? 


7.  Harry  is  6  years  old,  and  Frank  is  4  years  older 
than  Harry.  How  old  is  Frank  ?  ANS.  :  10  years  old : 
6  years  and  4  years  are  10  years. 


82  ORAL  LESSONS  IN  NUMBER 

Mary  is  9  years  old,  and  Susan  is  4  years  younger 
than  Mary.  How  old  is  Susan? 

There  are  8  peach  trees  and  4  pear  trees  in  a  garden. 
How  many  trees  in  the  garden  ?  How  many  more 
peach  trees  than  pear  trees? 

Samuel  caught  7  fishes  and  Henry  caught  4  fishes. 
How  many  fishes  did  both  boys  catch?  How  many 
fishes  did  Samuel  catch  more  than  Henry? 

A  wagon  contains  9  boys  and  4  girls.  How  many 
children  in  the  wagon?  If  4  of  the  boys  leave,  how 
many  children  will  be  left  in  the  wagon? 

Lucy  picked  3  bunches  of  pansies,  with  4  pansies  in 
each  bunch.  How  many  pansies  did  she  pick?  How 
many  4  pansies? 

Frank  has  8  cents,  and  he  wishes  to  buy  a  slate 
worth  12  cents.  How  many  more  cents  must  he  have 
to  buy  the  slate? 

Howard  picked  13  peaches  and  ate  4  of  them.  How 
many  peaches  had  he  left? 

Charles  paid  10  cents  for  a  slate  and  4  cents  for  a 
bunch  of  pencils.  How  much  did  he  pay  for  slate  and 
pencils  ?  How  many  cents  did  the  slate  cost  more  than 
the  pencils? 


BOARD   AND   SLATE    EXERCISES. 


abcdefgh          a  b  c  d  e 
44444444          42444 

Add  _2_jLAAjLJLj?.J*.          44442 

14321 
24234 

abcdefgh          Addji  J  _1  _2  _3 
From   6879    11    10    13    12 
Take    44444444 


SECOND- YEAR  COURSE.  83 


3 


\ 
\ 


\ 


/" 


8 4 4  12 4 

*     s  /     \ 

\  /x   '    N 

S7  lo 

6  10 


NOTE.— See  Lesson  III,  page  74,  notes. 


LESSON  VI. 

1.  There  are  2  men  in  one  wagon,  and  5  men  in 
another  wagon.  How  many  men  in  both  wagons? 

How  many  men  are  2  men  and  5  men?  1  man  and 
5  men  ?  3  men  and  5  men  ? 

How  many  are  1  and  5?    2  and  5?     3  and  5? 

There  are  7  men  in  a  wagon.  If  5  men  get  out,  how 
many  men  will  be  left  in  the  wagon? 

How  many  men  are  6  men  less  5  men?  7  men  less 
5  men?  8  men  less  5  men? 

How  many  are  6  less  5?    7  less  5?    8  less  5? 


2.  How  many  blocks  are  4  blocks  and  4  blocks?  8 
blocks  less  4  blocks  ?  5  blocks  and  4  blocks  ?  9  blocks 
less  4  blocks?  6  blocks  and  4  blocks?  10  blocks  less 

4  blocks? 

There  are  4  trees  in  a  row,  and  5  trees  in  another 
row.  How  many  trees  in  both  rows? 

How  many  trees  are  4  trees  and  5  trees?    5  trees  and 

5  trees?     6  trees  and  5  trees? 

How  many  are  4  and  5?    5  and  5?     6  and  5? 


84  ORAL  LESSONS  IN  NUMBER. 

A  man  brought  9  shade  trees  to  town,  and  sold  5  of 
them.  How  many  trees  had  he  left? 

How  many  trees  are  9  trees  less  5  trees?  10  trees  less 
5  trees?  11  trees  less  5  trees? 

How  many  are  9  less  5?     10  less  5?     11  less  5? 


3.  How  many  blocks  are  7  blocks  and  5  blocks?  12 
blocks  less  5  blocks?  8  blocks  and  5  blocks?  13 
blocks  less  5  blocks?  9  blocks  and  5  blocks?  14 
blocks  less  5  blocks? 

A  farmer  has  7  sheep  in  one  field,  and  5  sheep  in 
another  field.  How  many  sheep  in  both  fields? 

How  many  sheep  are  7  sheep  and  5  sheep?  8  sheep 
and  5  sheep?  9  sheep  and  5  sheep? 

How  many  are  7  and  5?     8  and  5?     9  and  5? 

A  farmer  has  12  sheep  :  if  he  sells  5  of  them,  how 
many  sheep  will  he  then  have? 

How  many  sheep  are  12  sheep  less  5  sheep?  13 
sheep  less  5  sheep?  14  sheep  less  5  sheep? 

How  many  are  12  less  5  ?     13  less  5  ?     14  less  5  ? 


4.  How  many  peaches  are  2  peaches  and  5  peaches? 
4  peaches  and  5  peaches  ?  6  peaches  and  5  peaches  ?  8 

peaches  and  5  peaches?  9  peaches  less  5  peaches? 

How  many  are: 

1  cent    and  5  cents  ?  6  cents  less  5  cents  ? 

2  cents  and  5  cents?  7  cents  less  5  cents? 

3  cents  and  5  cents?  8  cents  less  5  cents? 

4  cents  and  5  cents?  9  cents  less  5  cents? 

5  cents  and  5  cents?  10  cents  less  5  cents? 

6  cents  and  5  cents?  11  cents  less  5  cents? 

7  cents  and  5  cents?  12  cents  less  5  cents? 

8  cents  and  5  cents?  13  cents  less  5  cents? 

9  cents  and  5  cents?  14  cents  less  5  cents? 


SECOND-YEAR  COURSE. 


85 


5.  How  many  are :  Read  and  complete  : 

1+5=  6—5= 

2+5=  7—5= 

3+5=  8—5= 

4+5=  9—5= 

5  +  5=  10  —  5  = 

6  +  5=  11—5  = 

7  +  5=  12—5  = 

8  +  5=  13  —  5  = 

9  +  5=  14  —  5  = 

How  many  are  4  and  5  ?  9  less  5  ?  6  and  5  ?  11 
less  5?  8  and  5?  13  less  5?  3  and  5?  8  less  5?  5 
and  5?  10  less  5?  7  and  5?  12  less  5? 

How  many  are  3  and  5  ?  5  and  5  ?  7  and  5  ?  9  and 
5  ?  2  and  5  ?  4  and  5  ?  6  and  5  ?  8  and  5  ? 

How  many  are  8  less  5  ?  10  less  5  ?  12  less  5  ?  14 
less  5  ?  7  less  5  ?  9  less  5  ?  11  less  5  ?  13  less  5  ? 


How  many 

are: 

1  and  5  ? 

6  less  5? 

2  and  5? 

7  less  5  ? 

3  and  5  ? 

8  less  5? 

4  and  5  ? 

9  less  5? 

5  and  5  ? 

10  less  5  ? 

6  and  5  ? 

11  less  5? 

7  and  5  ? 

12  less  5? 

8  and  5? 

13  less  5  ? 

9  and  5  ? 

14  less  5  ? 

6.  Separate  each  of  the  numbers,  from  6  to  14  inclu- 
sive, into  5  and  another  number,  and  then  subtract  5 
and  its  complement  from  the  original  number,  thus : 


6 

is 

5 

and 

—  ,  or 

—  and 

5  /. 

6  less  5 

is  —  ; 

6  less 

—  is  5. 

7 

is 

5 

and 

—  or 

—  and 

5  .•; 

7  less  5 

is—  ; 

7  less 

—  is  5. 

8 

is 

5 

and 

—  or 

—  and 

5  .-. 

8  less  5 

is  —  ; 

8  less 

—  is  5. 

9 

is 

5 

and 

—  ,  or 

—  and 

5  /. 

9  less  5 

is—; 

9  less 

—  is  5. 

10 

is 

5 

and 

— 

/t 

10  less  5 

is  —  . 

11 

is 

5 

and 

—  or 

—  and 

5  /. 

11  less  5 

is  —  ; 

11  less 

—  is  5. 

12 

is 

5 

and 

—  ,  or 

—  and 

5  /. 

12  less  5  is  —  ; 

12  less 

—  is  5. 

13 

is 

5 

and 

—  or 

—  r  and 

5  .-. 

13  less  5 

is  —  ; 

13  less 

—  is  5. 

14 

is 

5 

and 

—  or 

—  and  5  .'. 

14  less  5 

is  —  ; 

14  less 

—  is  5. 

NOTE.— See  Lesson  III,  page  71,  notes. 


86  ORAL  LESSONS  IN  NUMBER. 

7.  How  many  balls  are  5  balls  and  5  balls?    5  balls 
and  5  balls  and  5  balls? 

How  many  balls  are  two  5  balls?     Three  5  balls? 

How  many  are  two  4's  ?    Three  4's  ? 

How  many  5  balls  in  10  balls? 

How  many  5  balls  in  15  balls? 

How  many  5's  in  10?     5's  in  15? 

How  many  5  pens  in  7  pens,  and  how  many  over? 

How  many  5  pens  in  11  pens,  and  how  many  over? 

How  many  5  pens  in  13  pens,  and  how  many  over? 

How  many  5  pens  in  12  pens,  and  how  many  over? 

How  many  5  pens  in  14  pens,  and  how  many  over? 

How  many  5's  in  15?    5's  in  10? 

How  many  3's  in  9?     3's  in  12?     3's  in  15? 

How  many  4's  in  8?     4's  in  12? 

How  many  2's  in  10?     2's  in  12?     2's  in  8? 


8.  A  hen  had  12  pretty  chickens,  and  a  hawk  carried 
off  5  of  them.  How  many  chickens  were  left? 

ANS.:  7  chickens :  12  chickens  less  5  chickens  are  7  chickens. 

There  are  8  windows  in  the  first  story  of  a  house,  and 
5  windows  in  the  second  story.  How  many  windows 
in  the  house?  How  many  more  windows  in  the  first 
story  than  in  the  second? 

Albert  is  9  years  old,  and  Frank  is  5  years  older  than 
Albert.  How  old  is  Frank? 

A  man  gave  13  dollars  for  a  watch,  and  5  dollars  for 
a  chain.  How  many  dollars  did  the  watch  cost  more 
than  the  chain  ? 

A  young  pear  tree  bore  12  pears,  and  5  of  them  were 
shaken  off  by  the  wind.  How  many  pears  were  left  on 
the  tree? 

Henry  paid  7  cents  for  an  inkstand,  and  5  cents  for 
a  pen.  How  many  cents  did  he  pay  for  both  ? 


SECOND-YEAR   COURSE. 


87 


There  are  10  bananas  in  one  bunch,  and  5  bananas  in 
another  bunch.  How  many  bananas  in  both  bunches? 
How  many  more  bananas  in  the  first  bunch  than  in 
the  second? 

Charles  gave  15  cents  for  a  knife,  and  5  cents  for  a 
comb.  How  many  cents  did  the  knife  cost  more  than 
the  comb? 

Kate  found  11  eggs  in  one  nest,  and  5  eggs  in  another 
nest.  How  many  eggs  in  the  first  nest  more  than  in 
the  second? 

There  are  three  rows  of  peach  trees  in  an  orchard, 
and  5  peach  trees  in  each  row.  How  many  peach  trees 
in  the  orchard? 


Add 


BOARD    AND   SLATE    EXERCISES. 


abed 


f  9 


55555555 

35246879 


abcdefgh 
From  8    10    7    9    11    13    12    14 
Take   55555555 


Add 


52535 
52423 
25332 
25223 
11152 


NOTE. — See  Lesson  III,  page  73,  notes. 


13— 


7  \  \ 


14' 


\ 


6  li 

NOTE. — See  Lesson  IV,  page  74,  notes. 


88  ORAL  LESSONS  IN  NUMBER. 

LESSON  VII. 
The  Numbers  16  to  20. 

NOTE. — The  object  of  this  lesson  is  to  develop  the  idea  of 
each  number  from  16  to  20  inclusive,  to  teach  its  name  and 
its  representation  by  figures.  See  Lesson  1,  page  67,  note. 

1.  A  few  weeks  ago,  children,  I  taught  you  the  num- 
bers from  10  to  15.  I  now  wish  to  teach  you  the  num- 
bers from  16  to  20.  We  shall  need  the  numeral  frame, 
a  package  of  10  pencils  and  10  loose  pencils,  and  20 
blocks. 

How  many  balls  on  this  wire?  "10  balls."  If  I 
put  2  balls  with  the  10  balls  (suiting  action  to  word), 
how  many  balls  will  there  be? 

10  balls  and  3  balls  are  how  many  balls?  10  balls 
and  4  balls  ?  10  balls  and  5  balls  ? 

Now,  if  I  put  6  balls  with  10  balls,  how  many  balls 
will  there  be?  "16  balls."  Right.  10  balls  and  6 
balls  are  16  balls. 

Here  are  10  pencils.  How  many  pencils  must  I  put 
with  them  to  make  16  pencils?  "6  pencils."  Right. 
10  pencils  and  6  pencils  are  how  many  pencils  ?  "  16 
pencils." 

I  now  make  10  lines  on  the  board.  How  many  more 
lines  must  I  add  to  make  16  lines?  "6  lines."  10 
lines  and  6  lines  are  how  many  lines? 

How  many  are  10  and  6? 

Here  are  the  figures  that  stand  for  sixteen:  26. 

What  does  the  1  stand  for?  "One  ten."  What  does 
the  6  stand  for?  "Six  ones." 

Make   16  lines  on  your  slate,  and   write 
16  below  them.  HUH 

Make  11,  12,  13,  14,  15,  16  on  your  slate.  16 


SECOND-YEAR  COURSE.  89 

2.  Here  are  10  balls  again.  If  I  add  7  balls  (suiting 
action  to  word),  how  many  balls  will  there  be?  (If  no 
pupil  can  tell,  the  teacher  should  give  the  answer.)  10 
balls  and  7  balls  are  17  balls. 

How  many  pencils  are  10  pencils  and  7  pencils? 
"17  pencils." 

How  many  lines  must  I  add  to  10  lines  to  make  17 
lines?  "  7  lines."  Right.  I  will  make  the  7  lines. 

How  many  are  10  lines  and  7  lines?  10  boys  and  7 
boys?  10  cents  and  7  cents?  10  and  7? 

Here  are  the  figures  that  stand  for  seventeen :  _/  7. 

What  does  the  1  stand  for?  What  does  the  7  stand 
for? 

Make  17  lines  on  your  slate,  and  write  17 
below  them. 

Make  11,  12,  13,  14,   15,  16,  17  on  your  17 

slate. 


3,  Here  are  10  balls,  and  how  many  balls  do  I  move 
beneath  them?  "8  balls."  10  balls  and  8  balls  are  18 
balls. 

How  many  balls  must  I  put  with  10  balls  to  make 
18  balls?  "8  balls." 

How  many  pencils  are  10  pencils  and  8  pencils? 
"  18  pencils." 

I  put  10  blocks  on  the  table,  and  Kate  may  put  with 
them  enough  blocks  to  make  18.  How  many  blocks 
has  she  put  on  the  table?  "8  blocks." 

10  blocks  and  8  blocks  are  how  many  blocks  ?  "  18 
blocks." 

10  and  8  are  how  many? 

Mary  may  put  10  blocks  on  the  table,  and  Kate  8 
blocks.  How  many  blocks  on  the  table? 

Kate  may  take  away  her  8  blocks.  How  many 
blocks  are  left  on  the  table?  "  10  blocks." 

0.  L.— 8. 


90  ORAL  LESSONS  IN  NUMBER. 

8  pears  taken  from  18  pears  leave  how  many  pears? 

Here  are  the  figures  that  stand  for  eighteen :  28. 

What  does  the  1  stand  for  ?     What  does  the  8  stand 
for? 

Make  18  lines  on  your  slate,   and  write 
18  below  them. 

Make  13,  14,  15,  16,  17,  18  on  your  slate.  18 


4,  Here  are  10  balls,  and  how  many  balls  do  I  move 
beneath  them?  "9  balls."  10  balls  and  9  balls  are  19 
balls. 

How  many  pencils  must  I  put  with  10  pencils  to 
make  19  pencils  ?  "  9  pencils."  10  pencils  and  9  pen- 
cils (suiting  action  to  word),  are  how  many  pencils? 

Jane  may  put  10  blocks  in  a  row  on  the  table,  and 
Samuel  may  put  9  blocks  in  another  row.  How  many 
blocks  are  10  blocks  and  9  blocks? 

How  many  boys  are  10  boys  and  9  boys?  How 
many  are  10  and  9? 

These  are  the  figures  that  stand  for  nineteen :  19. 

What  does  the  1  stand  for?  What  does  the  9  stand 
for? 

Make  19  lines  on  your  slate,  and  write 
19  below  them. 

Make  11,  12,  13,  14,  15,  16,  17,  18,  19  on  19 

your  slate. 


5.  We  have  at  last  reached  20  !  How  many  balls  on 
this  wire?  "10  balls."  How  many  balls  on  this  wire? 
"  10  balls."  How  many  10  balls  on  both  wires?  "Two 
10  balls."  Two  tens  are  called  twenty.  How  many 
balls  on  the  two  wires? 

How  many  pencils  in  this  bunch?  "10  pencils." 
Let  us  tie  10  more  pencils  in  a  second  bunch. 


SECOND-YEAR  COURSE.  91 

How  many  pencils  in  my  right  hand  ?  "  10  pencils." 
How  many  pencils  in  my  left  hand  ?  "  10  pencils."  How 
many  10  pencils  in  both  hands?  "Two  10  pencils." 

What  are  two  10's  called?  "Twenty."  Then  how 
many  pencils  in  both  my  hands? 

How  many  lines  are  10  lines  and  10  lines? 

How  many  blocks  are  10  blocks  and  10  blocks? 

Here  are  2  dimes.  How  many  cents  in  this  dime? 
"10  cents."  How  many  cents  in  this?  "10  cents." 
How  many  10  cents  in  two  dimes?  "Two  10  cents." 
How  many  cents  in  two  10  cents  ?  "  20  cents." 

How  many  cents  in  2  dimes?     "20  cents." 

These  are  the  figures  that  stand  for  twenty  :  %0. 

What  does  the  2  stand  for?  "  Two  10' s.  What  does 
the  0  stand  for?  "No  one."  Right,  and  so  20  stands 
for  two  10's  or  twenty. 

Make   20   lines    on    your   slate,    in    two 
groups  of  10  lines  each. 

Make  10  and  20  on  your  slate,  each  five  20 

times. 

6.  How  many  are  10  and  2?  10  and  4?  10  and  7? 
10  and  1?  10  and  3?  10  and  5?  10  and  8?  10  and 
9?  10  and  6?  10  and  10?  Two  10's? 

2  from  12  leaves  how  many  ?  3  from  13  ?  5  from  15  ? 
7  from  17?  9  from  19?  4  from  14?  6  from  16?  8 
from  18?  10  from  20? 


LESSON  VIII. 


1.  How  many  fingers  are  1  finger  and  6  fingers?  2 
fingers  and  6  fingers?  8  fingers  less  6  fingers?  3  fin- 
gers and  6  fingers?  9  fingers  less  6  fingers? 

There  are  2  baggage  cars  and  6  passenger  cars  in  a 
train.  How  many  cars  in  the  train? 


92  ORAL  LESSONS  IN  NUMBER. 

How  many  cars  are  2  cars  and  6  cars?  1  car  and  6 
cars?  3  cars  and  6  cars? 

How  many  are  1  and  6?     2  and  6?     3  and  6? 

There  are  8  kites  flying  in  the  air.  If  the  boys  haul 
down  6  kites,  how  many  kites  will  be  left  flying? 

How  many  kites  are  7  kites  less  6  kites  ?  8  kites  less 
6  kites?  9  kites  less  6  kites? 

How  many  are  7  less  6?    8  less  6?    9  less  6? 


2.  How  many  balls  are  4  balls  and  6  balls?  10  balls 
less  6  balls?  5  balls  and  6  balls?  11  balls  less  6 
balls?  6  balls  and  6  balls?  12  balls  less  6  balls? 

NOTE. — See  Lesson  V,  page  79,  note. 

There  are  4  trees  in  one  row  and  6  trees  in  another 
row.  How  many  trees  in  both  rows? 

How  many  trees  are  4  trees  and  6  trees?  5  trees  and 
6  trees?  6  trees  and  6  trees? 

How  many  are  4  and  6?     5  and  6?     6  and  6? 

Kate  picked  10  pansies,  and  gave  six  of  them  to  her 
mother.  How  many  pansies  had  Kate  left? 

How  many  pansies  are  10  pansies  less  6  pansies?  12 
pansies  less  6  pansies?  11  pansies  less  6  pansies? 

How  many  are  4  and  6?  10  less  6?  11  less  6?  12 
less  6?  8  less  6?  9  less  6?  7  less  6? 


3.  How  many  balls  are  7  balls  and  6  balls?  13  balls 
less  6  balls  ?  8  balls  and  6  balls  ?  14  balls  less  6  balls  ? 
9  balls  and  6  balls?  15  balls  less  6  balls? 

NOTE.— Use  the  numeral  frame,  and  number  the  large  groups 
by  "picking  out"  ten. 


SECOND-YEAR  COURSE.  93 

Jane  picked  7  roses  from  one  bush  and  6  roses  from 
another  bush.  How  many  roses  did  she  pick  from  both 
bushes  ? 

How  many  roses  are  7  roses  and  6  roses  ?  8  roses  and 
6  roses?  9  roses  and  6  roses? 

How  many  are  7  and  6?     8  and  6?    9  and  6? 

Jane  picked  13  roses,  and  gave  6  of  them  to  her 
sister.  How  many  roses  had  Jane  left? 

How  many  roses  are  13  roses  less  6  roses?  14  roses 
less  6  roses?  15  roses  less  6  roses? 

How  many  are  13  less  6?     14  less  6?     15  less  6? 


4.  How  many  oranges  are  5  oranges  and  6  oranges? 
11  oranges  less  6  oranges?  7  oranges  and  6  oranges? 
13  oranges  less  6  oranges?  9  oranges  and  6  oranges? 
15  oranges  less  6  oranges? 

How  many  peaches  are  4  peaches  and  6  peaches?  10 
peaches  less  6  peaches?  6  peaches  and  6  peaches?  12 
peaches  less  6  peaches  ?  8  peaches  and  6  peaches  ?  14 
peaches  less  6  peaches? 

How  many  are : 

2  lemons  and  6  lemons  ?  8  lemons  less  6  lemons  ? 

3  lemons  and  6  lemons  ?  9  lemons  less  6  lemons  ? 

4  lemons  and  6  lemons  ?  10  lemons  less  6  lemons  ? 

5  lemons  and  6  lemons?  11  lemons  less  6  lemons? 

6  lemons  and  6  lemons  ?  12  lemons  less  6  lemons  ? 

7  lemons  and  6  lemons  ?  13  lemons  less  6  lemons  ? 

8  lemons  and  6  lemons  ?  14  lemons  less  6  lemons  ? 

9  lemons  and  6  lemons  ?  5  lemons  less  6  lemons  ? 

How  many  pears  are  7  pears  and  6  pears?  9  pears 
and  6  pears?  8  pears  and  6  pears? 

How  many  pears  are  13  pears  less  6  pears?  15  pears 
less  6  pears?  14  pears  less  6  pears? 


94 


ORAL  LESSONS  IN  NUMBER. 


5.  How  many 

are  : 

Read  and 

1  and  6? 

7  less  6? 

1  +  6  = 

2  and  6? 

8  less  6? 

2  +  6  = 

3  and  6? 

•  9  less  6? 

3  +  6  = 

4  and  6? 

10  less  6? 

4  +  6  = 

5  and  6? 

11  less  6? 

5  +  6  = 

6  and  6  ? 

12  less  6? 

6  +  6  = 

7  and  6? 

13  less  6? 

7  +  6  = 

8  and  6? 

14  less  6? 

8  +  6  = 

9  and  6? 

15  less  6? 

9  +  6  = 

7  —  6: 

8  —  6 

9  —  6 

10  —  6 

11  —  6 

12  —  6 

13  —  6 

14  —  6 
15—6 


How  many  are  2  and  6?  8  less  6  ?  4  and  6  ?  10  less 
6?  6  and  6?  12  less  6?  8  and  6?  14  less  6? 

How  many  are  3  and  6  ?  9  less  6  ?  5  and  6  ?  11  less 
6?  7  and  6?  13  less  6?  9  and  6?  15  less  6? 

How  many  are  9  less  6?  11  less  6?  13  less  6?  15 
less  6?  14  less  6?  12  less  6?  10  less  6?  8  less  6? 


6.  Separate  each  number  from  7  to  15  into  6  and 
another  number,  and  then  subtract  6  and  its  comple- 
ment from  the  original  number,  thus: 


7  is  6  and  —  ,  or  —  and  6    .*. 
8  is  6  and  —  ,  or  —  and  6    /. 
9  is  6  and  —  ,  or  —  and  6   /. 
10  is  6  and  —  ,  or  —  and  6    .'. 
11  is  6  and  —  ,  or  —  and  6    .-. 
12  is  6  and—, 
13  is  6  and  —  ,  or  —  and  6    .'. 
14is6and  —  ,or  —  and  6    /. 
15  is  6  and  —  ,  or  —  and  6   /. 

7  less  6  is  —  ;    7  less  —  is  6. 
8  less  6  is  —  ;    8  less  —  is  6. 
9  less  6  is  —  ;    9  less  —  is  6. 
10  less  6  is  —  ;  10  less  —  is  6. 
11  less  6  is  —  ;  11  less  —  is  6. 
12  less  6  is—. 
13  less  6  is  —  ;  13  less  —  is  6. 
14  less  6  is  —  ;  14  less  —  is  6. 
15  less  6  is  —  ;  15  less  —  is  6. 

NOTE. — See  Lesson  III,  page  71,  notes  1  and  2. 


SECOND-YEAR  COURSE.  95 

7.  How  many  are  6  balls  and  6  balls? 

How  many  balls  are  two  6  balls? 

How  many  are  two  6's? 

How.  many  6  shells  in  12  shells? 

How  many  6's  in  12? 

How  many  6  tops  in  8  tops,  and  how  many  over? 

How  many  6  tops  in  10  tops,  and  how  many  over? 

How  many  6  tops  in  13  tops,  and  how  many  over? 

How  many  6  tops  in  15  tops,  and  how  many  over? 

How  many  6  tops  in  16  tops,  and  how  many  over? 

How  many  2's  in  4?    2's  in  8?    2's  in  12? 

How  many  3's  in  6?    3's  in  9?     3's  in  12? 

How  many  4's  in  8?    4's  in  12?    4's  in  16? 


8.  A  school  yard  contains  5  maple  trees  and  6  elm 
trees.  How  many  trees  in  the  yard? 

A  huckster  bought  8  turkeys  of  one  farmer,  and  6 
turkeys  of  another  farmer.  How  many  turkeys  did  he 
buy? 

How  many  more  turkeys  did  he  buy  of  the  first 
farmer  than  of  the  second? 

John  found  9  eggs  in  one  nest,  and  6  eggs  in  another 
nest.  How  many  eggs  did  he  find  in  both  nests? 

How  many  eggs  in  the  first  nest  more  than  in  the 
second  ? 

There  are  13  persons  in  a  street  car :  if  6  of  them 
leave  the  car,  how  many  will  be  left  in  it? 

Charles  is  11  years  old,  and  his  sister  is  6  years  old. 
How  much  older  is  Charles  than  his  sister? 

A  man  paid  8  dollars  for  a  pair  of  boots,  and  6  dol- 
lars for  a  hat.  How  much  did  he  pay  for  both  boots 
and  hat? 

How  much  did  he  pay  for  the  boots  more  than  for 
the  hat? 


96 


ORAL  LESSONS  IN  NUMBER. 


There  are  9  cherry  trees  in  a  row,  and  6  plum  trees 
in  another  row.  How  many  trees  in  the  two  rows? 
How  many  more  cherry  trees  than  plum  trees? 

How  many  radishes  in  2  bunches,  if  there  be  6 
radishes  in  each  bunch? 

BOARD   AND   SLATE   EXERCISES. 


a  b  c  d  e  f  g  h 
66666666 


Add 


abcdefgh 
From  9   10  8   11    13   12   14   15 

rjl  —  7*^.^?          /?£?          £?          /?          /*          /"*          /"* 
JL  /TA/6      DDDUDuOD 

NOTE. — See  Lesson  III,  page  73,  notes. 

2 

/ 


abode 
64563 
66426 
32342 
22233 
Add  12134 


9. 


8 6 4 

/ 

9X 


14- 6 -10 


15' 


,11 


13 


6  12 

NOTE.— See  Lesson  III,  page  74,  notes  1  and  2. 


LESSON  IX. 

1.  How  many  blocks  are  1  block  and  7  blocks?  8 
blocks  less  7  blocks  ?  2  blocks  and  7  blocks  ?  9  blocks 
less  7  blocks?  3  blocks  and  7  blocks?  10  blocks  less  7 
blocks? 


SECOND-YEAR  COURSE.  97 

How  many  boys  are  1  boy  and  7  boys?  2  boys  and 
7  boys?  3  boys  and  7  boys? 

How  many  are  1  and  7?    2  and  7?    3  and  7? 

A  boy  caught  10  fishes,  and  sold  7  of  them.  How 
many  fishes  had  he  left? 

How  many  fishes  are  10  fishes  less  3  fishes  ?  10  fishes 
less  2  fishes?  10  fishes  less  1  fish? 

How  many  are  8  less  7?    9  less  7?     10  less  7? 


2.  How  many  balls  are  4  balls  and  7  balls?  11  balls 
less  7  balls?  5  balls  and  7  balls?  12  balls  less  7  balls? 

6  balls  and  7  balls?     13  balls  less  7  balls? 

There  are  4  roses  on  one  bush,  and  7  roses  on  another 
bush.  How  many  roses  on  both  bushes? 

How  many  roses  are  4  roses  and  7  roses  ?    5  roses  and 

7  roses?    6  roses  and  7  roses? 

How  many  are  4  and  7?    5  and  7?     6  and  7? 

Clara  picked  12  roses,  and  gave  7  of  them  to  her 
mother?  How  many  roses  had  Clara  left? 

How  many  roses  are  12  roses  less  7  roses?  11  roses 
less  7  roses?  13  roses  less  7  roses? 

How  many  are  11  less  7?     12  less  7?     13  less  7? 


3.  How  many  balls  are  7  balls  and  7  balls?  14  balls 
less  7  balls?  8  balls  and  7  balls ?  15  balls  less  7  balls ? 
9  balls  and  7  balls?  16  balls  less  7  balls? 

NOTE. — See  Lesson  V,  page  79,  note. 

There  are  7  trees  in  one  row,  and  7  trees  in  another 
row.     How  many  trees  in  both  rows? 
How  many  trees  are  7  trees  and  7  trees  ?    8  trees  and 

7  trees?    9  trees  and  7  trees? 
o.  L  -9. 


98  ORAL  LESSONS  IN  NUMBER. 

How  many  are  7  and  7?     8  and  7?    9  and  7? 

A  man  brought  14  melons  to  market,  and  sold  7 
them.     How  many  melons  had  he  left? 

How  many  melons  are  14  melons  less  7  melons? 
melons  less  7  melons?     16  melons  less  7  melons? 

How  many  are  14  less  7?     15  less  7?     16  less  7? 


of 


15 


4.  How  many  are  : 

3  oranges  and  7  oranges  ?            10  oranges  less  7  oranges  ? 

5  oranges  and  7  oranges?            12  oranges  less  7  oranges? 

7  oranges  and  7  oranges?           14  oranges  less  7  oranges? 

9  oranges  and  7  oranges  ?            16  oranges  less  7  oranges  ? 

2  lemons  and  7  lemons  ?              9  lemons  less  7  lemons  ? 

4  lemons  and  7  lemons?           11  lemons  less  7  lemons? 

6  lemons  and  7  lemons  ?           13  lemons  less  7  lemons  ? 

8  lemons  and  7  lemons?            15  lemons  less  7  lemons? 

5,  How  many  are  : 

Read  and  complete  : 

land  7?          8  less  7? 

1  +  7=         8  —  7-= 

2  and  7?          9  less  7? 

2  +  7=         9  —  7  = 

3  and  7?        10  less  7? 

3  +  7=        10  —  7  = 

4  and  7?        11  less  7? 

4  +  7=        11  —  7  = 

5  and  7?        12  less  7? 

5  +  7=        12  —  7  = 

6  and  7?        13  less  7? 

6  +  7=        13—7  = 

7  and  7?        14  less  7? 

7  +  7=        14  —  7  = 

8  and  7?        15  less  7? 

8  +  7=        15  —  7  = 

9  and  7?        16  less  7? 

9  +  7=        16—7  = 

How  many  are  3  and  7?  10  less  7?  5  and  7?  12 
less  7?  7  and  7?  14  less  7?  9  and  7?  16  less  7? 

How  many  are  4  and  7?  11  less  7?  6  and  7?  13 
less  7?  8  and  7?  15  less  7?  2  and  7?  9  less  7? 


SECOND-YEAR   COURSE.  99 

How  many  are  10  less  7?  10  less  3?  12  less  7?  12 
less  3?  16  less  7?  16  less  9?  13  less  7?  13  less  6? 
15  less  7?  15  less  8? 

How  many  are  3  and  7  ?  5  and  7  ?  7  and  7  ?  9  and 
7?  2  and  7?  4  and  7?  6  and  7?  8  and  7? 

How  many  are  10  less  7?  12  less  7?  14  less  7?  16 
less  7?  9  less  7?  11  less  7?  13  less  7?  15  less  7? 


6.  Separate  each  number,  from  8  to  16  inclusive,  into 
7  and  another  number,  and  subtract  7  and  its  comple- 
ment, thus  found,  from  each  original  number,  thus: 

8  is  7  and  — ,  or  —  and  7  .'.  8  less  7  is  — ;     8  less  —  is  7. 

9  is  7  and  — ,  or  —  and  7  .'.    9  less  7  is  — ;    9  less  —  is  7. 

10  is  7  and  — ,  or  —  and  7  .'.  10  less  7  is  — ;  10  less  —  is  7. 

11  is  7  and  — ,  or  —  and  7  .'.  11  less  7  is  —  ;  11  less  —  is  7. 

12  is  7  and  — ,  or  —  and  7  .'.  12  less  7  is  — ;  12  less  —  is  7. 

13  is  7  and  — ,  or  —  and  7  .'.  13  less  7  is  — :  13  less  —  is  7. 

14  is  7  and  —  .'.14  less  7  is  — . 

15  is  7  and  — ,  or  —  and  7  /.  15  less  7  is  — ;  15  less  —  is  7. 

16  is  7  and  — ,  or  —  and  7  .'.  16  less  7  is  — ;  16  less  —  is  7. 

NOTE. — See  Lesson  III,  page  71,  notes. 


7.  How  many  balls  are  7  balls  and  7  balls,  or  two  7 
balls? 

How  many  7  balls  in  14  balls? 

How  many  7  balls  in  15  balls,  and  how  many  balls 
over? 

How  many  7  balls  in  10  balls,  and  how  many  balls 
over? 

How  many  7  balls  in  16  balls,  and  how  many  balls 
over? 


100  ORAL  LESSONS  IN  NUMBER. 

How  many  7  balls  in  12  balls,  and  how  many  balls 
over? 

How  many  4's  in  10,  and  how  many  ones  over? 
How  many  4's  in  14,  and  how  many  ones  over? 
How  many  4's  in  11,  and  how  many  ones  over? 
How  many  5's  in  17,  and  how  many  ones  over? 
How  many  5's  in  13,  and  how  many  ones  over? 
How  many  7's  in  15,  and  how  many  ones  over? 
How  many  7's  in  17,  and  how  many  ones  over? 


8.  Harry  is  6  years  old,  and  James  is  7  years  older 
than  Harry.  How  old  is  James? 

There  are  8  girls  and  7  boys  in  a  class.  How  many 
pupils  in  the  class? 

There  are  9  keys  in  one  bunch,  and  7  keys  in  another 
bunch.  How  many  keys  in  both  bunches? 

Harry  has  13  marbles,  and  Mark  has  7  marbles. 
How  many  more  marbles  has  Harry  than  Mark? 

There  are  7  ducks  in  the  water,  and  7  ducks  on  the 
land.  How  many  ducks  in  the  flock? 

A  hunter  brought  home  14  quails,  and  gave  7  of 
them  to  a  neighbor.'  How  many  quails  had  he  left? 

James  caught  5  fishes,  and  Clarence  caught  7  fishes. 
How  many  fishes  did  both  catch? 

A  little  boy  earned  15  cents  by  doing  errands,  and 
then  paid  7  cents  for  a  present  to  his  sister.  How 
many  cents  had  he  left? 

A  drover  bought  16  cows,  and  then  sold  7  of  them. 
How  many  cows  had  he  left? 

A  boy  picked  13  pears,  and  gave  7  of  them  to  his 
father.  How  many  pears  had  he  left? 

How  many  cherries  in  2  bunches,  of  7  cherries  each? 

Mary  picked  14  roses,  and  tied  them  in  bunches  of  7 
roses  each.  How  many  bunches  did  they  make? 


SECOND- YEAR   COURSE. 


101 


BOARD    AND   SLATE    EXERCISES. 


a    b     c     d    e     f    g     h 

77777777 
Add   24357869 


a  b     c    d 


f    g 


From  9  11  10  12  14  15  13  16      Add 
Take          T777 


a  b  c  d  e  f 
223123 
272143 
717737 
223532 
242432 
331231 


NOTE.  —  See  Lesson  III,  page  73,  notes. 

2  9 


r 


ft 


ia 


15 7 


N14 


NOTE. — See  Lesson  III,  page  74,  notes. 


LESSON  X. 

1.  How  many  shells  are  1  shell  and  8  shells?  9 
shells  less  8  shells?  2  shells  and  8  shells?  10  shells 
less  8  shells?  3  shells  and  8  shells?  11  shells  less  8 
shells? 

Willie  found  2  eggs  in  one  nest  and  8  eggs  in  another 
nest.  How  many  eggs  did  he  find? 

How  many  eggs  are  1  egg  and  8  eggs?  2  eggs  and  8 
eggs?  3  eggs  and  8  eggs? 


102  ORAL  LESSONS  IN  NUMBER 

How  many  are  1  and  8?    2  and  8?    3  and  8? 

A  man  bought  10  oranges,  and  gave  8  of  them  to  his 
children.  How  many  oranges  had  the  man  left? 

How  many  oranges  are  9  oranges  less  8  oranges?  10 
oranges  less  8  oranges?  11  oranges  less  8  oranges? 

How  many  are  9  less  8?     10  less  8?     11  less  8? 


2.  How  many  balls  are  4  balls  and  8  balls?  12  balls 
less  8  balls?  5  balls  and  8  balls?  13  balls  less  8  balls? 
6  balls  and  8  balls?  14  balls  less  8  balls? 

There  are  4  boys  and  8  girls  in  a  class.  How  many 
pupils  in  the  class? 

How  many  apples  are  4  apples  and  8  apples?  5 
apples  and  8  apples?  6  apples  and  8  apples? 

How  many  are  4  and  8?    5  and  8?     6  and  8? 

Mary  picked  12  plums,  and  gave  8  of  them  to  her 
mother.  How  many  plums  had  Mary  left? 

How  many  plums  are  4  plums  and  8  plums?  5 
plums  and  8  plums?  6  plums  and  8  plums? 

How  many  are  4  and  8?    5  and  8?    6  and  8? 


3.  How  many  balls  are  7  balls  and  8  balls?  15  balls 
less  8  balls?  8  balls  and  8  balls?  16  balls  less  8  balls? 
9  balls  and  8  balls?  17  balls  less  8  balls? 

NOTE. — See  Lesson  V,  page  79,  note. 

There  are  7  girls  in  one  class  and  8  girls  in  another 
class.  How  many  girls  in  both  classes? 

How  many  girls  are  7  girls  and  8  girls?  8  girls  and 
8  girls?  9  girls  and  8  girls? 

How  many  are  7  and  8?     8  and  8?    9  and  8? 


SECOND-YEAR  COURSE. 


103 


How  many  balls  are  left  when  8  balls  are  taken  from 
15  balls?  8  balls  from  16  balls?  8  balls  from  17 
balls? 

How  many  lemons  are  15  lemons  less  8  lemons?  16 
lemons  less  8  lemons?  17  lemons  less  8  lemons? 

How  many  are  15  less  8?     16  less  8?     17  less  8? 


4.  How  many  are : 

3  roses  and  8  roses?  11  roses 

5  roses  and  8  roses?  13  roses 

7  roses  and  8  roses?  15  roses 
9  roses  and  8  roses?  17  roses 
2  pears  and  8  pears?  10  pears 

4  pears  and  8  pears?  12  pears 

6  pears  and  8  pears?  14  pears 

8  pears  and  8  pears?  16  pears 


less  8 
less  8 
less  8 
less  8 
less  8 
less  8 
less  8 
less  8 


roses? 

roses? 

roses? 

roses? 

pears? 

pears? 

pears? 

pears? 


How  many  are  9  cents  less  8  cents?  11  cents  less  8 
cents?  13  cents  less  8  cents?  15  cents  less  8  cents? 
17  cents  less  8  cents? 

How  many  are  10  cents  less  8  cents?  12  cents  less  8 
cents?  14  cents  less  8  cents?  16  cents  less  8  cents? 


How  many 

are  : 

Read  and  complete: 

1  and  8? 

9  less  8? 

1  +  8  = 

9  —  8 

2  and  8? 

10  less  8? 

2  +  8  = 

10  —  8 

3  and  8  ? 

11  less  8? 

3  +  8  = 

11  —  8 

4  and  8? 

12  less  8? 

4  +  8  = 

12  —  8 

5  and  8? 

13  less  8? 

5  +  8  = 

13  —  8 

6  and  8  ? 

14  less  8? 

6  +  8  = 

14  —  8 

7  and  8? 

15  less  8? 

7  +  8  = 

15  —  8 

8  and  8  ? 

16  less  8? 

8  +  8  = 

16  —  8 

9  and  8  ? 

17  less  8? 

9  +  8  = 

17  —  8 

104  ORAL  LESSONS  IN  NUMBER. 

How  many  are  1  and  8?  3  and  8?  5  and  8?  7 
and  8?  9  and  8? 

How  many  are  2  and  8?  4  and  8?  6  and  8?  8 
and  8?  10  and  8? 

How  many  are  3  and  8?  11  less  8?  13  less  8?  15 
less  8?  17  less  8?  10  less  8?  12  less  8?  14  less  8? 
16  less  8? 


7.  Separate  each  of  the  numbers,  from  9  to  17  inclu- 
sive, into  8  and  another  number,  and  then  subtract  8 
and  its  complement  from  the  original  number,  thus  : 

9  is  8  and  — ,  or  —  and  8  .'.    9  less  8  is  — ;     9  less  —  is  8. 

10  is  8  and  — ,  or  —  and  8  /.  10  less  8  is  — ;  10  less  —  is  8. 

11  is  8  and  — ,  or  —  and  8  .*.  11  less  8  is  —  ;  11  less  —  is  8. 

12  is  8  and  — ,  or  —  and  8  /.  12  less  8  is  — ;  12  less  —  is  8. 

13  is  8  and  — ,  or  —  and  8  .'.  13  less  8  is  — ;  13  less  —  is  8. 

14  is  8  and  — ,  or  —  and  8  .'.  14  less  8  is  — ;  14  less  —  is  8. 

15  is  8  and  — ,  or  —  and  8  /.  15  less  8  is  — ;  15  less  —  is  8. 

16  is  8  and  -  .-.  16  less  8  is  — . 

17  is  8  and  — ,  or  —  and  8  /.  17  less  8  is  —  ;  17  less  —  is  8. 

NOTE. — See  Lesson  III,  page  71,  notes. 


8.  How  many  balls  are  8  and  8  balls,  or  two  8  balls? 
How  many  8  balls  in  16  balls? 
How  many  are  two  8's? 
How  many  8's  in  16? 

How  many  8  balls  in  10  balls,  and  how  many  over? 
How  many  8  balls  in  12  balls,  and  how  many  over? 
How  many  8  balls  in  17  balls,  and  how  many  over? 
How  many  8  balls  in  18  balls,  and  how  many  over? 
How  many  4's  in  8?  4's  in  12?  4's  in  16? 


SECOND-YEAR  COURSE.  105 

How  many  5's  in  10?    5's  in  15?    5's  in  20? 
How  many  5's  in  17,  and  how  many  over? 
How  many  5's  in  14,  and  how  many  over? 


9.  A  boy  caught  2  fishes,  and  sold  1  of  them  for  5 
cents  and  the  other  for  8  cents.  How  many  cents  did 
he  receive? 

ANS.  :    13  cents:  5  cents  and  8  cents  are  13  cents. 

Harry  bought  a  box  of  figs,  and  gave  8  figs  to  his 
brother,  and  had  8  figs  left.  How  many  figs  did  he 
buy? 

A  boy  was  carrying  a  basket  with  15  eggs  in  it,  and 
a  large  dog  ran  against  the  basket  and  broke  8  of  the 
eggs.  How  many  eggs  were  left  unbroken? 

Clarence  is  14  years  old  and  Edgar  is  8  years  old. 
How  much  older  is  Clarence  than  Edgar? 

Susan  wrote  18  words  on  her  slate,  and  then  rubbed 
out  8  words.  How  many  words  were  left  on  her  slate? 

Frank  caught  17  fine  trout,  and  sold  8  of  them  to 
Edward.  How  many  trout  had  Frank  left? 

A  grocer  bought  a  dozen  brooms,  and  sold  8  of  them. 
How  many  brooms  had  he  left? 

Edward's  father  gave  him  10  cents,  and  his  mother 
gave  him  5  cents.  How  many  cents  had  he? 

If  Edward  should  pay  8  cents  for  a  slate,  how  many 
cents  would  he  have  left? 

How  many  more  cents  in  12  cents  than  in  8  cents? 
In  15  cents  than  in  8  cents? 

Charles  has  18  cents,  and  William  has  8  cents.  How 
many  more  cents  has  Charles  than  William? 

Helen  has  17  roses,  and  Kate  has  8  roses.  How  many 
many  more  roses  has  Helen  than  Kate? 

John  has  9  marbles,  and  James  8  marbles.  How 
many  marbles  have  both  of  the  boys? 


106 


ORAL  LESSONS  IN  NUMBER. 


BOARD    AND    SLATE    EXERCISES. 


abcdefghi 

888888888 


a  b  c  d  e  f  g 
1121233 

Add  .lAAAAAA^Ll.  2322332 

4221623 
abcdefghi  8888445 

FromW  9  11  13  12  14  16  15  17     Add  S_J>_  JLZJ*J>J> 

Take    888888888 


NOTE. — See  Lesson  III,  page  73,  notes. 


10 


4  16 8 12 


4 


NOTE. — See  Lesson  III,  page  74,  notes. 


X15 


14 


LESSON  XI. 

1,  How  many  shells  are  1  shell  and  9  shells?  10 
shells  less  9  shells?  2  shells  and  9  shells?  11  shells 
less  9  shells?  3  shells  and  9  shells?  12  shells  less  9 
shells? 

There  are  9  sheep  in  one  field,  and  3  sheep  in  another 
field.  How  many  sheep  in  both  fields? 

How  many  sheep  are  3  sheep  and  9  sheep?  2  sheep 
and  9  sheep?  1  sheep  and  9  sheep? 

How  many  are  1  and  9?    2  and  9?    3  and  9? 


SECOND-YEAR  COURSE.  107 

Kate  wrote  12  words  on  her  slate,  and  then  rubbed 
out  9  of  them.  How  many  words  were  left  on  her 
slate  ? 

How  many  words  are  10  words  less  9  words?  11 
words  less  9  words?  12  words  less  9  words? 

How  many  are  10  less  9?     11  less  9?     12  less  9? 


2.  How  many  balls  are  4  balls  and  9  balls?  13  balls 
less  9  balls?  5  balls  and  9  balls?  14  balls  less  9  balls ? 
6  balls  and  9  balls?  15  balls  less  9  balls? 

NOTE. — See  Lesson  V,  page   79,  note. 

There  are  4  boys  and  9  girls  in  a  reading  class.  How 
many  pupils  in  the  class? 

How  many  pupils  are  4  pupils  and  9  pupils?  5 
pupils  and  9  pupils?  6  pupils  and  9  pupils? 

How  many  are  4  and  9?    5  and  9?     6  and  9? 

There  are  13  pupils  in  a  class,  and  9  of  them  are 
girls.  How  many  boys  in  the  class? 

How  many  pupils  are  13  pupils  less  9  pupils?  14 
pupils  less  9  pupils?  15  pupils  less  9  pupils? 

How  many  are  13  less  9?     14  less  9?     15  less  9? 


3.  How  many  balls  are  7  balls  and  9  balls?  16  balls 
less  9  balls  ?  8  balls  and  9  balls  ?  17  balls  less  9  balls  ? 
9  balls  and  9  balls?  18  balls  less  9  balls? 

John  has  7  marbles  in  his  left  hand  and  9  marbles  in 
his  right  hand.  How  many  marbles  has  John  in  both 
hands  ? 

How  many  marbles  are  7  marbles  and  9  marbles?  8 
marbles  and  9  marbles?  9  marbles  and  9  marbles? 

How  many  are  7  and  9?    8  and  9?    9  and  9? 

Charles  picked  16  cherries,  and  gave  9  of  them  to  his 
sister.  How  many  cherries  had  he  left? 


108 


ORAL  LESSONS  IN  NUMBER. 


How  many  cherries  are  16  cherries  less  9  cherries? 
17  cherries  less  9  cherries?  18  cherries  less  9  cherries? 

How  many  are  16  less  9?     17  less  9?     18  less  9? 

How  many  are  3  men  and  9  men  ?  5  men  and  9 
men?  7  men  and  9  men?  9  men  and  9  men? 

How  many  are  2  sheep  and  9  sheep?  4  sheep  and  9 
sheep?  6  sheep  and  9  sheep?  8  sheep  and  9  sheep? 


4.  How  many  are: 

2  figs  and  9  figs? 

4  figs  and  9  figs? 

6  figs  and  9  figs? 

8  figs  and  9  figs? 

1  cent  and  9  cents? 

3  cents  and  9  cents? 

5  cents  and  9  cents? 

7  cents  and  9  cents? 

9  cents  and  9  cents? 


11  figs  less  9 

13  figs  less  9 

15  figs  less  9 

17  figs  less  9 

10  cents  less 

12  cents  less 

14  cents  less 

16  cents  less 

18  cents  less 


figs? 
figs? 
figs? 
figs? 

9  cents? 
9  cents? 
9  cents? 
9  cents? 
9  cents? 


5.  How  many  are: 

Read  and  complete: 

land  9?        10  less  9? 

1  +  9=        10  —  9 

2  and  9?         11  less  9? 

2  +  9=        11—9 

3  and  9  ?         12  less  9  ? 

3  +  9=        12  —  9 

4  and  9?        13  less  9? 

4  +  9=        13  —  9 

5  and  9?        14  less  9? 

5  +  9=        14  —  9 

6  and  9?         15  less  9? 

6  +  9=        15  —  9 

7  and  9?        16  less  9? 

7-1-9=        16  —  9 

8  and  9?         17  less  9? 

8  +  9=        17  —  9 

9  and  9?        18  less  9? 

9  +  9=        18  —  9 

How  many  are  2  and  9?  4  and  9?  6  and  9?  8 
and  9?  3  and  9?  5  and  9?  7  and  9?  9  and  9? 

How  many  are  11  less  9?  13  less  9?  15  less  9?  17 
less  9  ?  12  less  9  ?  14  less  9  ?  16  less  9  ?  18  less  9  ? 


SECOND-YEAR  COURSE.  109 

6.  Separate  each  of  the  numbers,  from  10  to  18  inclu- 
sive, into  9  and  another  number,  and  then  subtract  9 
and  its  complement,  thus  found,  from  the  original 
numbers,  thus: 

10  is  9  and  — ,  or  —  and  9  /.  10  less  9  is  — ;  10  less  —  is  9. 

11  is  9  and  — ,  or  —  and  9  /.  11  less  9  is  — ;  11  less  —  is  9. 

12  is  9  and  — ,  or  —  and  9  /.  12  less  9  is  — ;  12  less  —  is  9. 

13  is  9  and  — ,  or  —  and  9  /.  13  less  9  is  —  ;  13  less  —  is  9. 

14  is  9  and  — ,  or  —  and  9  /.  14  less  9  is  — ;  14  less  —  is  9. 

15  is  9  and  — ,  or  —  and  9  /.  15  less  9  is  — ;  15  less  —  is  9. 

16  is  9  and  — ,  or  —  and  9  /.  16  less  9  is  — ;  16  less  —  is  9. 

17  is  9  and  — ,  or  —  and  9  .*.  17  less  9  is  — ;  17  less  —  is  9. 

18  is  9  and  -  /.  18  less  9  is  — . 

NOTE.— See  Lesson  III,  page  71,  notes. 


7.  How  many  balls  are  9  balls  and  9  balls,  or  two  9 
balls? 

How  many  are  two  9's? 

How  many  9  balls  in  18  balls? 

How  many  9's  in  18? 

How  many  9  balls  in  12  balls,  and  how  many  over? 

How  many  9  balls  in  15  balls,  and  how  many  over? 

How  many  9  balls  in  20  balls,  and  how  many  over? 

How  many  9  balls  in  19  balls,  and  how  many  over? 

How  many  6's  in  12?     6's  in  18? 

How  many  6's  in  16,  and  how  many  over? 

How  many  7's  in  16,  and  how  many  over? 

How  many  5's  in  16,  and  how  many  over? 

How  many  4's  in  15,  and  how  many  over? 


8.  There  are  5  cows  in  one  field  and  9  cows  in 
another  field.  How  many  cows  in  both  fields?  How 
many  more  cows  in  the  second  field  than  in  the  first? 


110  ORAL  LESSONS  IN  NUMBER. 

Jane  read  7  verses,  and  Kate  read  9  verses  more  than 
Jane.  How  many  verses  did  Kate  read? 

A  farmer  has  13  acres  of  wheat  and  9  acres  of  corn. 
How  many  more  acres  of  wheat  than  of  corn  has  he? 

There  are  15  yards  of  ribbon  on  a  spool :  if  a  clerk 
sell  9  yards  of  the  ribbon,  how  many  yards  will  be  left 
on  the  spool? 

A  huckster  bought  8  chickens  of  one  farmer  and  7 
chickens  of  another.  How  many  chickens  did  he  buy  ? 
If  he  sell  9  chickens,  how  many  chickens  will  he  then 
have? 

A  man  started  to  walk  14  miles  to  reach  a  railroad 
station,  and  when  he  had  walked  9  miles  he  stopped  at 
a  hotel  for  dinner.  How  many  miles  was  he  from  the 
station  ? 

A  hunter  saw  15  quails  in  a  flock,  and  9  of  them 
flew  away.  How  many  quails  were  left? 

A  lady  gave  16  dollars  for  a  shawl  and  9  dollars  for  a 
dress.  How  many  dollars  did  the  shawl  cost  more 
than  the  dress? 

A  boy  earned  5  cents  on  Thursday,  4  cents  on  Fri- 
day, and  9  cents  on  Saturday.  How  many  cents  did 
he  earn  in  the  three  days? 

SLATE   AND   BOARD   EXERCISES. 

abcdefghi  a    b   c   d   e  f 

999999999  232112 

Add  i_i-i_2A-^-*L.L9       222221 

319999 
993333 

abcdefghi        Add  L±jLJL_lJ> 
From    10  12  14  11  13  15  17  16  18 
Take     _9_  j)  _9^  _9^  J)  J)  J)  J)    9_ 

NOTE. — See  Lesson  III,  page  73,  notes. 


SECOND-YEAR  COURSE. 


Ill 


11 


3\ 

/5            la\ 

/4 

V 

/                                 \ 

\ 
\ 

/                                    \ 

/ 

\ 

/                                         \ 

/ 

8  j 

,  4              17  f 

)  13 

/ 

\                                        / 

\ 

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s 

^^                                  / 

\ 
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N16 

6 


NOTE.  —  See  Lesson  III,  page  74,  notes. 


LESSON  XII. 

Primary  Combinations  in  Addition. 

1.  There  are  only  forty-five  primary  combinations  in 
addition,  and  these  are  produced  by  adding  each  dig- 
ital number  (10  not  included)  to  itself  and  to  each  of 
the  higher  digital  numbers,  as  follows : 

a  b          c          d          e          f          g         h          i 

1+1  2+2  3+3  4+4  5+5  6+6  7+7  8+8  9+9 

2+1  3+2  4+3  5+4  6+5  7+6  8+7  9+8 

3+1  4+2  5+3  6+4  7+5  8+6  9+7 

4+1  5+2  6+3  7+4  8+5  9+6 

5+1  6+2  7+3  8+4  9+5 

6+1  7+2  8+3  9+4 

74-1  8+2  9+3 

8+1  9+2 

9+1 

NOTE. — The  above  combinations  should  be  written  on  the 
board,  and  the  pupils  drilled  in  adding  each  from  left  to  right, 
and  from  right  to  left  (the  first  in  each  column  excepted), 
thus:  "b"  =  3  +  2  =  5;  2  +  3  =  5.  The  two  may  be  combined, 
thus:  3  +  2  or  2  +  3  =  5.  This  is  a  condensed  addition  table, 


112  ORAL  LESSONS  IN  NUMBER. 

2.  These  primary  combinations  in  addition,  and  the 
inverse  processes  in  subtraction,  may  be  written  on  the 
board,  as  follows  : 


1  +  1  =  2 

2  —  1  =  ] 

2  +  1  or  1  +  2  = 

3  —  1  = 

;  3-2 

3  +  1  or  1  +  3  = 

4  ^  

;  4-3 

4  +  1  or  1+4  = 

5  —  1  = 

;  5-4 

5  +  1  or  1  +  5  = 

6  —  1  = 

;  6-5 

6  +  1  or  1  +  6  = 

7  —  1  = 

;  7-6 

7  +  1  or  1  +  7  = 

8  —  1  = 

;  8-7 

8  +  1  or  1  +  8  = 

9  —  1  = 

;  9-8 

9  +  1  or  1  +  9  = 

.-.   10  —  1  = 

;  10  —  9 

2  +  2=4  /.   4  —  2  =  2 

3  +  2  or  2  +  3=  .v   5  —  2=  ;  5  —  3  = 

4  +  2  or  2+4=  /.    6  —  2=  ;  6  —  4  = 

5  +  2  or  2  +  5=  /.    7  —  2=  ;  7  —  5  = 

6  +  2  or  2  +  6=  /.    8  —  2=  ;  8  —  6  = 

7  +  2  or  2  +  7=  /.    9  —  2=  ;  9  —  7  = 

8  +  2  or  2  +  8=  .-.  10  —  2=  ;10  —  8  = 

9  +  2  or  2  +  9=  /.  11  —  2=  ;11—  9  = 

3  +  3=  /.    6  —  3  = 

4  +  3  or  3  +  4=  /.    7  —  3=  ;  7  —  4  = 

5  +  3  or  3  +  5=  .-.    8  —  3=  ;  8  —  5  = 
6+3  or  3  +  6=  A    9  —  3=  ;  9  —  6  = 

7  +  3or3  +  7=  A  10  —  3=  ;10  —  7  = 

8  +  3  or  3  +  8=  .'.  11  —  3=  ;11  —  8  = 

9  +  3  or  3  +  9=  /.  12  —  3=  ;12  —  9  = 

4  +  4=  '/.    8  —  4  = 

5+4  or  4  +  5=  /.    9—4=  ;  9  —  5  = 

6  +  4  or  4+6=  /.  10—4=  ;10  —  6  = 

7  +  4  or  4  +  7==  ...  11  —  4=  ;11  —  7  = 

8  +  4  or  4  +  8=  .'.  12  —  4=  ;12  —  8  = 

9  +  4  or  4  +  9=  /.  13  —  4=  ;  13  —  9  = 


SECOND- YEAR   COURSE.  113 


5  +  5  = 

/.   10  —  5  = 

6  +  5  or  5  +  6  = 

/.   11  —  5  = 

;  11  —  6: 

7  +  5  or  5  +  7  = 

/.   12  —  5  = 

;  12-7: 

8  +  5  or  5  +  8  = 

.-.   13-5  = 

;  13  —  8: 

9  +  5  or  5  +  9  = 

.-.   14  —  5  = 

;  14  —  9: 

6  +  6  = 

/.   12  —  6  = 

7  +  6  or  6  +  7  = 

/.   13  —  6  = 

;  13  —  7: 

8  +  6  or  6  +  8  = 

.-.   14—6  = 

;  14  —  8: 

9+6  or  6  +  9  = 

/.   15  —  6  = 

;  15—9 

7  +  7  = 

/.   14  —  7  = 

8  +  7  or  7  +  8  = 

.-.   15  —  7  = 

;  15  —  8: 

9  +  7  or  7  +  9  = 

/.   16  —  7  = 

;  16  —  9: 

8  +  8  = 

/.   16  —  8  = 

9  +  8  or  8  +  9  = 

.-.   17  —  8  = 

;  17  —  9 

9  +  9  = 

/.   18  —  9  = 

LESSON  XIII. 

Separate  each  number,  from  11  to  19  inclusive,  into 
two  digital  numbers  (including  10),  and  take  from  it 
each  digital  number  found,  thus: 

11  is  10  and  — ,  or  1  and  — .'.  11  less  1  is  — ;  11  less  10  is  — . 
11  is  9  and  — ,  or  2  and  —  .'.11  less  2  is  —  ;  11  less  9  is  — . 
11  is  8 and — , or  3  and  —  /.  11  less  3  is — ;  11  less  Sis — . 
11  is  7  and — ,  or  4  and — .'.11  less  4  is  — ;  11  less  7  is — . 
His  6and — ,or5and  —  /.  11  less  5  is — ;  11  less  6is — . 


12  is  10  and  — ,  or  2  and  —  .  •.  12  less  2  is  —  ;  12  less  10  is  — . 
12  is  9  and — ,  or  3  and  —  .'.  12  less  3  is  —  ;  12  less  9  is — . 
12  is  8  and  — ,  or  4  and  —  /.  12  less  4  is  — ;  12  less  8  is  — . 
12  is  7  and  — ,  or  5  and  —  /.  12  less  5  is  — ;  12  less  7  is  — . 
12  is  6  and—,  .\121ess6is—. 


o.  L.-IO. 


114 


ORAL   LESSONS  IN  NUMBER. 


13  is  1 0  and  — ,  or  3  and  — 
13  is  9  and  — ,  or  4  and  — 
13  is  Sand — , or 5 and  — 


13  less  3  is  — ;  13  less  10  is  — 
13 less 4 is  — ;  13 less  9 is—. 
13  less 5  is  — ;  13 less  Sis— . 


13  is    7  and  — ,  or  6 and—  /.  13  less  6 is—  ;  13 less   7  is—. 


14  is  10  and  — ,  or  4  and 
14  is  9  and  — ,  or  5  and 
14  is  8  and  — ,  or  6  and 
14 is  7 and—, 


14  less  4  is — ;  14  less  10  is 
14  less  5  is  —  ;  14  less  9  is 
14 less 6 is  — ;  14 less  Sis 
14  less  7  is  — . 


15  is  10  and  — ,  or  5  and 
15  is  9  and — ,  or  6  and 
15  is  8  and  —  or  7  and 


15  less  5  is  —  ;  15  less  10  is 
15  less  6  is  —  ;  15  less  9  is 
15  less  7  is  —  ;  15  less  8  is 


16  is  10  and  — ,  or  6  and 
16  is  9  and — ,  or  7  and 
16 is  Sand—, 


17  is  10  and  — ,  or  7  and 
17  is   9  and  — ,  or  8  and 


16  less  6  is  — ;  16  less  10  is 
16 less 7 is— ;  16 less  9 is 
16  less  8  is—. 


17  less  7  is— ;  17  less  10  is 
17  less  8  is  — ;  17  less   9  is 


18  is  10  and  — ,  or  8  and 
18  is    9  and  — , 


18  less  8  is  — ;  18  less  10  is 
18  less  9  is—. 


19  is  10  and  — ,  or  9  and  —  /.  19  less  9  is  —  ;  19  less  10  is  — . 

NOTE. — The  above  exercises  may  be  written  on  the  board, 
using  the  signs  --)-,  — ,  and  =,  respectively,  in  place  of  "and," 
"less,"  and  "is."  They  may  also  be  copied  by  the  pupils  on 
their  slates,  and  the  blank  spaces  filled  with  the  proper  num- 
bers. 


THIRD-YEAR    COURSE. 


NOTE. — The  first  fourteen  lessons  in  the  ELEMENTARY  ARITH- 
METIC (now  used  by  the  pupils)  are  chiefly  devoted  to  addition 
and  subtraction ;  and,  since  the  needed  preparatory  oral  train- 
ing is  given  in  the  second-year  course,  but  little  oral  instruction 
will  be  required  the  first  half  of  the  third  school  year. 

The  exercises  in  the  first  lesson  (page  116)  are  introductory  to 
the  third  year  lessons  in  addition  and  subtraction,  as  well  as 
to  those  in  multiplication  and  division,  and  hence  the  exercises 
in  this  first  lesson  should  precede  Lesson  III  in  the  ELEMENTARY 
ARITHMETIC.  The  "Supplemental  Blackboard  Exercises"  in 
addition  and  subtraction  given  in  this  manual  (on  page  155), 
should  accompany  the  blackboard  exercises  in  the  ELEMENTARY 
ARITHMETIC. 

With  the  above  exceptions,  the  following  lessons  are  intended 
to  be  introductory  to  and  to  accompany  the  exercises  in  multi- 
plication and  division  in  the  ELEMENTARY  ARITHMETIC,  and 
hence  may  be  given  the  last  half  of  the  third  school  year.  It 
is  suggested  that  the  exercises  in  this  manual  precede  the  cor- 
responding exercises  in  the  book  used  by  the  pupils. 

In  some  schools,  it  may  be  found  practicable  and  desirable 
to  give  a  part  of  this  oral  course  in  multiplication  and  division 
in  the  latter  part  of  the  second  school  year,  before  an  elementary 
book  is  put  into  the  hands  of  the  pupils.  When  this  is  done, 
no  attempt  should  be  made  to  advance  beyond  Lesson  V. 

ORAL  LESSONS  IN  MULTIPLICATION  AND  DIVISION. 

Aims. 

To  teach  (1)  the  product  of  any  two  digital  numbers,  and 
(2)  the  division  of  this  product  by  each  of  its  two  factors. 

(115) 


116  ORAL  LESSONS  IN  NUMBER. 


Steps. 

1.  The  finding  of  the  number  corresponding  to  the 
product  of  any  two  digital  numbers,  by  adding  one  of 
them  to  itself  continuously  as  many  times  as  there  are 
units  in   the  other  given  number,   less  one;   or  (2)  by 
adding  one  of  the  numbers  to  its  product  by  a  number 
one  less  than  the  other  given  number. 

2.  The  fixing  of  the  product  of  any  two  digital  num- 
bers  in   the   memory,   so   that   it   may   be   recalled   in- 
stantly, without  adding,  when  its  two  factors  are  given. 

3.  The  teaching  of  the  division  of  any  product  by 
each  of  its  two  digital  factors  as  the  inverse  of  the  pro- 
cess of  their  multiplication. 


LESSON  I. 

The  Numbers  20  to  100. 

NOTE. — The  object  of  this  lesson  is  to  develop  the  numbers 
from  20  to  100  inclusive,  teach  their  names,  and  their  repre- 
sentation by  figures. 

The  teacher  should  be  supplied  with  a  numeral  frame,  ten 
bunches  of  rods  or  cards  of  ten  each,  and  ten  loose  rods  or 
cards ;  also  a  blackboard. 

In  asking  the  questions,  the  teacher  should  present  the  ap- 
propriate objects,  and  otherwise  "  suit  the  action  to  the  word." 


1.  Here  is  the  numeral  frame.  Let  us  see  how  many 
balls  on  it. 

How  many  wires  in  the  numeral  frame  ?     "  10  wires." 

How  many  balls  on  the  first  wire?  On  the  second 
wire  ?  On  each  wire  ?  "  10  balls." 

How  many  10  balls  on  the  two  upper  wires  ?  "  Two 
10  balls."  How  many  balls  on  these  two  wires? 

Here  are  the  figures  that  stand  for  twenty:  20. 


THIRD- YEAR  COURSE.  117 

How  many  10  balls  on  the  three  upper  wires? 
"Three  10  balls."  How  many  balls?  "30  balls." 
Three  tens  are  how  many? 

Here  are  the  figures  that  stand  for  thirty :  $0. 

How  many  10  balls  on  the  four  upper  wires  ?  "  Four 
10  balls."  Four  10  balls  are  how  many  balls? 

How  many  balls  on  any  4  wires?  Four  tens  are  how 
many? 

Here  are  the  figures  that  stand  for  forty  :  ^Q. 

How  many  10  balls  on  the  5  upper  wires?  Five  10 
balls  are  how  many  balls? 

How  many  balls  on  any  5  wires?  Five  tens  are  how 
many? 

Here  are  the  figures  that  stand  for  fifty  :  SO. 

NOTE. — In  like  manner,  teach  the  numbers  60,  70,  80,  90,  100. 

There  are  10  rods  (or  cards)  in  each  of  these  bunches. 
How  many  rods  in  2  bunches  ?  3  bunches  ?  5  bunches  ? 
6  bunches?  4  bunches?  7  bunches?  9  bunches?  8 
bunches?  10  bunches? 

NOTE.— Write  on  the  blackboard  the  numbers  10,  20,  30,  40, 
50,  60,  70,  80,  90,  100 ;  and,  pointing  to  them  promiscuously,  have 
the  pupils  read  them.  Next  call  attention  to  the  fact  that  the 
figures  1,  2,  3,  etc.,  denote  the  number  of  tens,  40  being  4  tens, 
50  being  5  tens,  etc. 

Write  neatly  on  your  slates  the  numbers  10,  20,  30, 
etc.,  to  100. 


2.  How  many  balls  on  the  two  upper  wires?  I  will 
now  slide  these  20  balls  to  the  right  end  of  the  wires, 
and  all  the  other  balls  to  the  left  end  of  the  wires. 

If  I  now  slide  1  ball  on  the  third  wire  to  the  right 
under  the  20  balls,  how  many  balls  will  there  be  at  the 
right?  "21  balls." 


118  ORAL  LESSONS  IN  NUMBER. 

If  I  slide  2  balls  on  the  third  wire  under  the  20 
balls,  how  many  balls  will  there  be? 

NOTE.— Slide  back  the  1  ball  and  slide  2  balls  under  the  20 
balls.  Slide  successively  3  balls,  4  balls,  etc.,  under  the  20  balls, 
and  have  pupils  give  the  number  of  balls.  Then  ask  the  fol- 
lowing questions,  sliding  in  each  case  the  designated  number  of 
balls  under  the  20  balls. 

How  many  balls  are  20  balls  and  1  ball?  20  balls 
and  2  balls?  20  balls  and  3  balls?  20  balls  and  4 
balls?  20  balls  and  5  balls?  20  balls  and  6  balls?  20 
balls  and  7  balls?  20  balls  and  8  balls?  20  balls  and 
9  balls?  20  balls  and  10  balls? 

NOTE. — Take  the  pencils  (or  cards),  and  holding  2  bunches  in 
your  left  hand,  add  successively  2  pencils,  5  pencils,  3  pencils, 
etc.,  and  have  the  pupils  give  in  each  case  the  number  of  pencils. 


3.  How  many  are : 

20  and  1?  20  and     6? 

20  and  2?  20  and     7? 

20  and  3?  20  and     8? 

20  and  4?  20  and     9? 

20  and  5?  20  and  10? 

I  will  now  write  on  the  board  a  line  of  ten  2's,  thus : 
2222222222 

What  figure  nAist  I  make  at  the  right  of  the  first  2 
to  express  twenty?  "0."  At  the  right  of  the  second  2 
to  express  twenty-one ?  At  the  right  of  the  third  2  to 
express  twenty-two  f  At  the  right  of  the  fourth  2  to  ex- 
express  twenty-three? 

NOTE. — In  like  manner,  write  the  figures  4,  5,  etc.,  at  the  right 
of  the  remaining  2's  in  the  line,  and  teach  what  number  the 
two  figures  in  each  case  express. 


THIRD-YEAR  COURSE.  119 

What  number  do  I  now  write  on  the  board?  (Write 
23  on  the  board.)  How  many  does  the  figure  2  de- 
note ?  "2  tens  or  twenty."  How  many  does  the  figure 
3  denote?  "3  ones." 

NOTE. — In  like  manner,  teach  what  each  figure  in  24,  25,  etc., 
expresses.  Write  on  the  blackboard  promiscuously  the  numbers 
from  20  to  29  inclusive,  and,  pointing  to  them,  have  the  pupils 
read  the  numbers  designated. 

Write  neatly  on  your  slates,  in  a  column,  all  the 
numbers  from  20  to  29  inclusive. 

NOTE. — In  like  manner,  teach  the  numbers  between  30  and 
40,  and  between  40  and  50.  When  this  is  done,  the  pupils  will 
be  able  to  write  the  numbers  between  50  and  60,  60  and  70,  70 
and  80,  80  and  90,  and  90  and  100, 


LESSON  II. 

1,  How  many  times  do  I  say  la,  children :  La,  la. 
How  many  times  now :  La,  la,  la.  Say  la  twice.  Say 
la  three  times.  Say  la  four  times. 

Clap  your  hands  twice.  Clap  them  3  times.  4  times. 
5  times. 

How  many  times  do  I  write  the  figure  2  on  the 
board?  (Write  it  3  times  in  one  group,  4  times  in 
another,  etc.) 

How  many  times  1  finger  do  I  hold  up?  (Hold  up 
3  fingers,  4  fingers,  etc.) 

How  many  pencils  are  3  times  1  pencil?  4  times  1 
pencil?  6  times  1  pencil? 

How  many  are  three  1's?  3  times  1?  Five  1's?  5 
times  1?  Seven  1's?  7  times  1? 


120  ORAL  LESSONS  IN  NUMBER. 

2.  How  many  are: 
2  +  2,  or  2  twos? 
2  +  2  +  2,  or  3  twos? 
2  +  2  +  2  +  2,  or  4  twos? 
2  +  2  +  2  +  2  +  2,  or  5  twos? 
2  +  2  +  2  +  2  +  2  +  2,  or  6  twos  ? 
2  +  2  +  2  +  2  +  2  +  2  +  2,  or  7  twos? 
2  +  2  +  2  +  2  +  2  +  2  +  2  +  2,  or  8  twos? 
2  +  2  +  2  +  2  +  2  +  2  +  2  +  2  +  2,  or  9  twos? 
.    2  +  2+2  +  2  +  2  +  2  +  2  +  2  +  2  +  2,  or  10  twos? 


3.  How  many  are: 

Two  2  balls?  2  times  2  balls? 

Three  2  balls?  3  times  2  balls? 

Four  2  balls?  4  times  2  balls? 

Five  2  balls?  5  times  2  balls? 

Six  2  balls?  6  times  2  balls? 

Seven  2  balls?  7  times  2  balls? 

Eight  2  balls?  8  times  2  balls? 

Nine  2  balls?  9  times  2  balls? 

Ten  2  balls?  10  times  2  balls? 

NOTE. — Take  the  numeral  frame  and  slide  all  the  balls  to  the 
left-hand  side,  and  slide  to  the  right-hand  side  2  balls  on  the 
first  wire,  2  balls  on  the  second  wire,  2  balls  on  the  third  wire, 
etc.,  to  correspond  with  the  question.  If  in  any  case  the  pupil 
can  not  give  the  product,  let  him  add  2  balls  to  the  product 
next  preceding.  Continue  the  drill,  asking  the  questions  pro- 
miscuously, and  sliding  the  balls  in  a  corresponding  manner. 

How  many  blocks  are  2  times  2  blocks?  3  times  2 
blocks  ?  5  times  2  blocks  ?  7  times  2  blocks  ?  9  times 
2  blocks? 

How  many  blocks  are  4  times  2  blocks?  6  times  2 
blocks?  8  times  2  blocks?  10  times  2  blocks? 


THIRD- YEAR  COURSE. 


121 


4.  How  many  are: 

3  times  2  horses  ? 

5  times  2  horses? 
2  times  2  bushels  ? 

6  times  2  bushels? 

4  times  2  yards? 

8  times  2  yards? 

7  times  2  yards  ? 

9  times  2  dollars? 
10  times  2  dollars? 


How  many  times  : 
2  horses  in  6  horses? 
2  horses  in  10  horses  ? 
2  bushels  in  4  bushels? 
2  bushels  in  6  bushels  ? 
2  yards  in  8  yards? 
2  yards  in  16  yards  ? 
2  yards  in  14  yards? 
2  dollars  in  18  dollars  ? 
2  dollars  in  20  dollars? 


1  time    2? 

2  times  2? 

3  times  2? 

4  times  2? 

5  times  2? 

6  times  2? 

7  times  2? 

8  times  2? 

9  times  2? 


2  in 
2  in 
2  in 
2  in 


2? 

4? 

6? 

8? 
2  in  10? 
2  in  12? 
2  in  14? 
2  in  16? 
2  in  18? 


NOTES. — 1.  Write  the  above  tables  on  the  board,  and  have 
them  recited  in  three  ways,  as  follows :  First,  The  two  tables  to- 
gether, thus  (taking  third  line) :  3  times  2  are  6,  2  in  6  three 
times.  Second,  The  two  tables  separately,  drilling  on  the  table 
of  products  until  they  are  recited  without  the  least  hesitation. 
Third,  The  tables  promiscuously. 

2.  There  are  advantages  in  forming  and  learning  the  tables 
of  products  in  consecutive  order,  but,  in  determining  an  unknown 
product,  pupils  should  be  required  to  add  the  multiplicand  to 
the  product  next  preceding. 

How  many  are  4  times  2  ?  6  times  2  ?  3  times  2  ? 
5  times  2?  7  times  2?  9  times  2?  8  times  2?  10 
times  2? 

How  many  times  2  in  8?  2  in  12?  2  in  6?  2  in 
10?  2  in  14?  2  in  18?  2  in  16?  2  in  20? 

O.  L.-ll. 


122 


ORAL  LESSONS  IN  NUMBER. 


6.  There  are  2  pints  in  a  quart :  how  many  pints  in 
5  quarts? 

ANS.  :  10  pints:    5  times  2  pints  are  10  pints. 

How  many  pints  in  4  quarts?     In   6  quarts?     In  8 
quarts?     In  10  quarts?     In  7  quarts? 

How  much  will  3  oranges  cost  at  2  cents  apiece?    7 
oranges?     9  oranges? 

What  is  the  cost  of  4  two-cent  postage  stamps?     6 
two-cent  stamps?     10  two-cent  stamps? 

There  are  2  horses  in  a  span :  how  many  horses  in  3 
spans?    5  spans?     7  spans? 

SLATE   AND   BOARD   EXERCISES. 


Copy  and 

fill: 

Copy  and  fill  : 

1X2  = 

2- 

-2  = 

3X2  = 

6-2 

2X2-= 

4- 

-2  = 

5X2  = 

10  —  2 

3X2  = 

6- 

-2  = 

6X2  = 

12—2 

4X2  = 

8- 

Q  

8X2  = 

16-2 

5X2  = 

10- 

-2  = 

4X2  = 

8  —  2 

6X2  = 

12- 

-2  = 

7X2  = 

14-^2 

7X2  = 

14- 

-2  = 

9X2  = 

18^-2 

8X2  = 

16- 

-2  = 

2X2  = 

4-2 

9X2  = 

18- 

-2  = 

7X2  = 

14  —  2 

10X2  = 

20- 

-2  = 

9X2  = 

18  —  2 

NOTE.— Read  the  sign  X  times,  and  the  sign  -s-,  divided  by. 


a        b        c        d        e         f        g        h 

22222222 

Multiply    !_!__6_3_^_9        5        8 


abed          e  f         9          h 

Divide  2)4  '  2)8    2)12    2)6    2)14    2)18    2)10    2)16 


THIRD- YEAR  COURSE. 


123 


The  following  "  Wheel  Exercises "  may  be  used   to 
supplement  the  above  board  drills: 


\!/5 

3 2 4 


f 


iar 


P 


16 2 8 


12 


NOTE.— In  the  left-hand  table  the  number  2,  at  the  center,  is 
to  be  multiplied  by  each  of  the  outer  numbers;  in  the  right- 
hand  table,  each  of  the  outer  numbers  is  to  be  divided  by  the 
number  2  at  the  center. 


LESSON  III. 


1.  How  many  balls  are: 
Two      3  balls? 
Three   3  balls? 
3  balls? 
3 


Four 

Five 

Six 

Seven 

Eight 

Nine 

Ten 


balls? 
3  balls? 
3  balls? 
3  balls? 
3  balls? 
3  balls? 


2  times  3 

3  times  3 

4  times  3 

5  times  3 

6  times  3 

7  times  3 

8  times  3 

9  times  3 
10  times  3 


balls? 
balls? 
balls? 
balls? 
balls? 
balls? 
balls? 
balls? 
balls  ? 


NOTE. — Slide  all  the  balls  to  the  left-hand  side  of  the  numeral 
frame,  as  in  Lesson  II,  and  then  slide  to  right-hand  side  as 
many  3  balls  as  are  designated  in  each  question.  Find  each 
unknown  product  by  adding  3  balls  to  the  product  next  pre- 
ceding. 


124 


ORAL  LESSONS  IN  NUMBER. 


I.  How  many  are  : 

3  times  3  peaches? 

5  times  3  peaches? 

7  times  3  pears? 
9  times  3  pears? 

2  times  3  melons? 

4  times  3  melons? 

6  times  3  dollars  ? 

8  times  3  dollars? 
10  times  3  dollars  ? 


How  many  times: 
3  peaches  in  9  peaches? 
3  peaches  in  15  peaches? 
3  pears  in  21  pears? 
3  pears  in  27  pears? 
3  melons  in  6  melons? 
3  melons  in  12  melons  ? 
3  dollars  in  18  dollars  ? 
3  dollars  in  24  dollars? 
3  dollars  in  30  dollars? 


1  time    3? 

3  in    3? 

2  times  3? 

3  in     6? 

3  times  3? 

3  in     9? 

4  times  3? 

3  in  12? 

5  times  3? 

3  in  15? 

6  times  3? 

3  in  18? 

7  times  3? 

3  in  21? 

8  times  3? 

3  in  24? 

9  times  3? 

3  in  27? 

10  times  3  ? 

3  in  30? 

NOTE. — See  Lesson  II,  page  121,  notes. 

How  many  are  5  times  3?    7  times  3?    9  times  3? 

6  times  3?    8  times  3?    10  times  3? 

How  many  are  4  times  3?    3  in  12?    6  times  3?    3 
in  18?    8  times  3?    3  in  24?     10  times  3?    3  in  30? 

7  times  3?    3  in  21?    9  times  3?    3  in  27? 


4,  There  are  3  feet  in  a  yard :  how  many  feet  in  4 
yards?  In  6  yards?  8  yards? 

A  stem  of  clover  has  3  leaves  :  how  many  leaves  on 
5  stems?  4  stems?  9  stems? 


THIRD-YEAR  COURSE. 


125 


John  walks  3  miles  a  day  in  going  to  school :  how 
many  miles  does  he  walk  in  5  days  ?  In  10  days  ? 

How  much  will  5  lemons  cost  at  3  cents  apiece?  8 
lemons  at  3  cents  apiece? 

How  many  lemons,  at  3  cents  each,  can  be  bought 
for  15  cents?  For  24  cents? 

How  much  will  6  pencils  cost  at  3  cents  each?  10 
pencils  ? 

How  many  pencils,  at  3  cents  each,  can  be  bought  for 
18  cents?  For  30  cents? 

There  are  3  bushels  of  wheat  in  a  sack:  how  many 
bushels  in  10  sacks?  In  7  sacks? 

Kate  is  3  years  old,  and  her  father  is  9  times  as  old 
as  she:  how  old  is  her  father? 

A  mason  earns  3  dollars  a  day  :  how  much  will  he 
earn  in  6  days?  In  10  days? 

How  many  times  3  dollars  in  15  dollars?  In  30  dol- 
lars? In  18  dollars? 


5. 


6=  2X 
9=  3X 
12  =  4  X 
15=  5  X 
18=  6  X 
21=  7  X 
24=  8  X 
27=  9  X 
30=10X 


,  or  3X 

/.      6- 

-3  = 

/.      9- 

-3  = 

,or  3X 

/.     12  - 

0    

,  or  3  X 

/.     15  - 

q  

,  or  3  X 

.%     18  - 

-3  = 

,or  3  X 

/.    21  - 

q  

,  or  3X 

/.     24- 

-3  = 

,or  3X 

/.    27- 

-3  = 

,or  3X 

/.    30- 

-3  = 

2  == 


;  12  -r-   4 : 
;  15-5-  5: 
;  18-4-  6 
;  21 -H-  7 
;  24--  8 
;  27^-  9 
;  30-^10 


NOTE.— Write  the  above  inverse  tables  on  the  blackboard, 
and  have  the  pupils  recite  them,  thus:  (3)  12  is  3  times  4  or 
4  times  3  /.  4  in  12  three  times;  3  in  12  four  times.  Or  (if 
preferred) :  12  equals  3  times  4  or  4  times  3  .*.  12  divided  by 
4  equals  3;  12  divided  by  3  equals  4.  Finally,  have  the  pupils 
copy  the  tables  neatly  on  their  slates,  filling  the  blanks. 


126 


ORAL  LESSONS  IN  NUMBER. 


SLATE    AND    BOARD    EXERCISES. 


1X3  = 

3-r-3 

2X3  = 

6-^-3 

3X3  = 

9-^3 

4X3  = 

12  --3 

5X3  = 

15  -=-3 

6X3  = 

18--3 

7X3  = 

21  --3 

8X3  = 

24  --3 

9X3  = 

27-^-3 

10X3  = 

30^3 

a 

b        c 

3 

3        3 

Multiply     3 

2        4 

2X3=  6  —  3: 

4X3=  12  —  3: 

6X3=  18  —  3: 

8X3=  24  —  3: 

3X3=  9  —  3 

5X3=  15  —  3 

7X3=  21—3 

9x3=  27  —  3: 

1X3=  3—3 

10X3=  30--3 


abed  e  f          g          h 

Divide   3)9     3)6     3)12     3)18     3)24     3)15     3)21     3)27 


3, 

,$                       9 

15 

X 

/                              ^ 

s 

\ 

\ 

s                                                             ^N 

s 

s 

N 

f                                                                                             N 

s 

8  : 

5  4             24  -NJ 

j'  12 

s' 

s 
' 

S\                                                                   s' 

s 

\ 

/ 

\               w'' 

\1 

18 


LESSON  IV. 

1.  How  many  balls  are: 

Two     4  balls?  2  times  4  balls? 

Three  4  balls?  3  times  4  balls? 

Four     4  balls?  4  times  4  balls? 


THIRD- YEAR  COURSE. 


127 


Five 

Six 

Seven 

Eight 

Nine 

Ten 


balls? 
balls? 
balls? 
balls? 
balls? 
balls? 


5  times  4 

6  times  4 

7  times  4 

8  times  4 

9  times  4 
10  times  4 


balls? 
balls? 
balls? 
balls? 
balls? 
balls? 


NOTE.— See  Lessbn  II,  page  120,  note. 


.  How  many  are : 

2  times  4  balls? 

4  times  4  balls? 

6  times  4  balls? 

3  times  4  balls? 

5  times  4  balls? 

7  times  4  balls? 
9  times  4  balls? 

8  times  4  balls? 
10  times  4  balls? 


How  many 
4  balls  in 
4  balls  in 
4  balls  in 
4  balls  in 
4  balls  in 
4  balls  in 
4  balls  in 
4  balls  in 
4  balls  in 


times : 
8  balls? 
16  balls? 
24  balls? 
12  balls? 
20  balls? 
28  balls? 
36  balls? 
32  balls? 
40  balls? 


3.         1  time    4? 

2  times  4? 

3  times  4? 

4  times  4? 

5  times  4? 

6  times  4? 

7  times  4? 

8  times  4? 

9  times  4? 
10  times  4  ? 

NOTE. — See  Lesson  II,  page  121,  notes. 


4  in  4? 
4  in  8? 
4  in  12? 
4  in  16? 
4  in  20? 
4  in  24? 
4  in  28? 
4  in  32? 
4  in  36? 
4  in  40? 


How  many  are  3  times  4?  5  times  4?  7  times  4? 
4  times  4?  6  times  4?  8  times  4?  7  times  4?  9 
times  4?  10  times  4? 


128  ORAL  LESSONS  IN  NUMBER. 

How  many  are  2  times  4?  4  in  8?  4  times  4?  4 
in  16?  6  times  4?  4  in  24?  3  times  4?  4  in  12?  5 
times  4?  4  in  20?  7  times  4?  4  in  28?  9  times  4? 
4  in  36?  8  times  4?  4  in  32? 


4.  There  are  4  quarts  in  a  gallon  :  how  many  quarts 
in  2  gallons?  In  4  gallons? 

There  are  4  pecks  in  a  bushel :  how  many  pecks  in  3 
bushels?  In  5  bushels? 

How  many  pecks  in  6  bushels?    In  8  bushels? 

A  horse  has  4  hoofs :  how  many  hoofs  have  10 
horses?  7  horses?  9  horses? 

How  many  shoes  does  it  take  to  shoe  a  horse?  To 
shoe  4  horses? 

How  much  will  6  lemons  cost  at  4  cents  apiece?  10 
lemons  at  4  cents  apiece? 

At  4  cents  apiece,  how  many  lemons  can  be  bought 
for  12  cents?  For  24  cents? 

Kate  bought  8  spools  of  thread  at  4  cents  a  spool : 
how  much  did  they  cost? 

How  many  spools  of  thread,  at  4  cents  each,  can  be 
bought  for  20  cents?  For  32  cents? 

What  will  6  bunches  of  beets  cost  at  4  cents  a 
bunch  ? 

How  many  bunches  of  beets,  at  4  cents  each,  can  be 
bought  for  24  cents?  For  28  cents? 

How  many  beets  in  5  bunches,  if  there  be  4  beets  in 
each  bunch? 

What  will  6  pencils  cost  at  4  cents  apiece? 

At  4  cents  apiece,  how  many  pencils  can  be  bought 
for  24  cents? 

At  4  cents  apiece,  how  many  oranges  can  be  bought 
for  28  cents? 

What  will  10  oranges  cost  at  4  cents  apiece? 


THIRD- YEAR   COURSE. 


129 


4=   IX      ,  or  4X       .'. 

4  —  4=     ;    4—    1  = 

8  =  2  X      ,  or  4  X 

8  —  4=     ;    8—  2  = 

12  =   3  X  '   ,  or  4  X       /. 

12  —  4=     ;  12—   3  = 

16=-   4X 

16-4  = 

20=  5  X      ,  or  4X       /. 

20  —  4=     ;  20—  5  = 

24  =   6X      ,  or  4X       / 

24  —  4=     ;  24—   6  = 

28=   7  X      ,  or  4X 

28  —  4=     ;  28—   7  = 

32  =   8  X     ,  or  4  X       /. 

32-4=     ;  32—   8  = 

36  =  9  X      ,  or  4  X       /. 

36  —  4=     ;  36-5-  9  = 

40  =  10X      ,or4X       / 

40-5-4  =     ;  40-5-10  = 

NOTE.—  See  Lesson  III,  page 

125,  note. 

SLATE    AND   BOARD   EXERCISES. 

1X4=    ;      4-4  = 

3X4=         12-5-4 

2x4=    ;      8-4  = 

5x4=         20  -5-  4 

3X4=    ;    12-4  = 

7X4=         28-5-4 

4x4=    ;    16-4  = 

4X4=         16-5-4 

5X4=    ;    20  —  4  = 

2x4=           8-4-4 

6X4=    ;    24-4  = 

6X4=         24-4-4 

7X4=    ;    28-4  = 

8X4=         32-4-4 

8X4=    ;    32  —  4  = 

.7X4=         28-4-4 

9X4=    ;    36-4  = 

9X4=     ;   36-4 

10x4=    ;    40-5-4  = 

1X4=     ;      4-5-4 

a        b        c 

d        e        f        g        h 

Multiply    444 

44444 

264 

5        3        9        7       J^ 

a        b          c 

d         e         f         g 

Divide    4)8    4)24    4)16    4)20    4)12    4)36    4)28 

130  ORAL  LESSONS  IN  NUMBER. 


2 

3N  ,5  12X 

\  /  \ 

\    '   / 

8 4 4  32 N  4  '- 16 


\ 


9'  N7  36' 


S28 
24 


LESSON  V. 

1.  How  many  balls  are : 

Two      5  balls?  2  times  5  balls? 

Three    5  balls?  3  times  5  balls? 

Four     5  balls?  4  times  5  balls? 

Five     5  balls?  5  times  5  balls? 

Six       5'  balls?  6  times  5  balls? 

Seven  5  balls?  7  times  5  balls? 

Eight    5  balls?  8  times  5  balls? 

Nine     5  balls?  9  times  5  balls? 

Ten      5  balls?  10  times  5  balls? 

NOTE.— See  Lesson  II,  page  120,  note. 


2.  How  many  are  :  How  many  times : 

3  times  5  chairs?  5  chairs  in  15  chairs? 

5  times  5  chairs?  5  chairs  in  25  chairs? 
2  times  5  books?  5  books  in  10  books? 

4  times  5  books?  5  books  in  20  books? 

6  times  5  pens?  5  pens  in  30  pens? 

8  times  5  pens?  5  pens  in  40  pens? 

7  times  5  cents?  5  cents  in  35  cents? 

9  times  5  cents?  5  cents  in  45  cents? 
10  times  5  cents?  5  cents  in  50  cents? 


THIRD- YE  A  E  COURSE.  131 

3.  1  time    5?  5  in    5? 

2  times  5?  5  in  10? 

3  times  5?  5  in  15? 

4  times  5?  5  in  20? 

5  times  5  ?  5  in  25  ? 

6  times  5?  5  in  30? 

7  times  5  ?  5  in  35  ? 

8  times  5?  5  in  40? 

9  times  5?  5  in  45? 
10  times  5?  5  in  50? 

NOTE. — See  Lesson  II,  page  121,  notes. 

How  many  are  5  times  5  ?  4  times  5  ?  6  times  5  ? 
8  times  5?  10  times  5?  7  times  5?  9  times  5?  3 
times  5? 

How  many  are  3  times  5  ?  5  in  15  ?  5  times  5  ?  5 
in  25  ?  7  times  5  ?  5  in  35  ?  9  times  5  ?  5  in  45  ?  4 
times  5?  5  in  20?  6  times  5?  5  in  30?  8  times  5? 
5  in  40?  10  times  5?  5  in  50? 


4.  There  are  5  nails  on  a  man's  hand :  how  many 
nails  on  2  hands?  On  4  hands? 

There  are  5  desks  in  a  row :  how  many  desks  in  4 
rows?  In  6  rows? 

There  are  5  cents  in  a  "  nickel " :  how  many  cents  in 
5  nickels?  In  10  nickels? 

If  there  be  5  radishes  in  a  bunch,  how  many  radishes 
in  7  bunches?  In  9  bunches? 

How  much  will  6  oranges  cost,  at  5  cents  apiece?  8 
oranges  at  5  cents  apiece? 

How  many  oranges,  at  5  cents  each,  can  be  bought 
for  30  cents?  For  40  cents? 

How  much  will  4  yards  of  ribbon  cost  at  5  cents  a 
yard?  8  yards  at  5  cents  a  yard? 


132 


ORAL  LESSONS  IN  NUMBER. 


How  many  yards  of  ribbon,  at  5  cents  a  yard,  can 
be  bought  for  40  cents?  For  50  cents? 

A  boy  put  35  beets  into  bunches  of  5  beets  each : 
how  many  bunches  did  they  make? 


5.    5  = 

1 

X 

,  or  5 

X     /.      5- 

-5  = 

5- 



1 

= 

10=== 

2 

X 

,  or  5 

X     .'.     10- 

-5  = 

10- 

2 

= 

15  = 

3 

X 

,  or  5 

X     .-.     15- 

-5  = 

15- 

3 

= 

20  = 

4 

X 

,  or  5 

X     .'.    20- 

-5  = 

20- 

4 

= 

25  = 

5 

X 

> 

/.     25  - 

-5  = 

30  = 

6 

X 

,  or  5 

X     .>     30- 

-5  = 

30- 

6  = 

35  = 

7 

X 

,  or  5 

X     /.     35- 

-5  = 

35- 

7 

rrr: 

40  = 

8 

X 

,  or  5 

X     A    40- 

-5  = 

40- 

8 

= 

45  = 

9 

X 

,  or  5 

X     /.    45- 

-5=    ; 

45- 

9 

= 

50  = 

10 

X 

,  or  5 

X     .%    50- 

-5=    ; 

50- 

10 

— 

NOTE.—  See 

Lesson  III, 

page  125,  note. 

BOARD 

AND   SLATE    EXERCISES. 

1X5 

— 

5- 

-5 

SB 

3X5  = 

15- 

-5  = 

2X5  = 

10- 

-5 

= 

1X5  = 

5- 

-5 

= 

3X5 

=s 

15- 

-5 

= 

4X5  = 

20- 

-5 

= 

4X5 

= 

20- 

-5 

= 

6X5  = 

30- 

-5 

= 

5X5 

= 

25- 

-5  = 

5X5  = 

25- 

-5 

= 

6X5  = 

30- 

-5 

= 

7X5  = 

35- 

-5 

= 

7X5 

= 

35- 

-5 

= 

2X5  = 

10- 

-5  = 

8X5 

= 

40- 

-5 

— 

8X5  = 

40- 

-5 

= 

9X5 

= 

45- 

-5 

— 

10X5  = 

50- 

-5 

= 

10X5 

= 

50- 

r-5 

*= 

9X5  = 

45- 

-5 

== 

a 

b 

c          d          e         f 

9 

h 

5 

5 

5555 

5 

5 

rultiply 

4 

3 

2869 

5 

7 

abed          e          f        g         h 
Divide   5)20    5)15    5)10    5)40    5)30    5)45    5)25    5)35 


\ 


THIRD- YEAR  COURSE. 
2  10 


133 


8 5 4  40 5 20 


35 


30 


LESSON 

VI. 

L  How  many  balls  are  : 

Two      6  balls? 

2  times  6  balls? 

Three  6  balls? 

3  times  6  balls? 

Four    6  balls? 

4  times  6  balls? 

Five     6  balls? 

5  times  6  balls? 

Six       6  balls? 

6  times  6  balls? 

Seven  6  balls? 

7  times  6  balls? 

Eight   6  balls? 

8  times  6  balls? 

Nine     6  balls? 

9  times  6  balls? 

Ten      6  balls? 

10  times  6  balls? 

NOTE. — See  Lesson  II,  page  120,  note. 


\.  How  many  are : 

3  times  6  boys? 

5  times  6  boys? 
2  times  6  slates? 

4  times  6  slates? 

6  times  6  knives  ? 

8  times  6  knives? 

7  times  6  words? 

9  times  6  words? 
10  times  6  cents? 


How  many  times: 
6  boys  in  18  boys? 
6  boys  in  30  boys  ? 
6  slates  in  12  slates? 
6  slates  in  24  slates  ? 
6  knives  in  36  knives? 
6  knives  in  48  knives? 
6  words  in  42  words? 
6  words  in  54  words? 
6  cents  in  60  cents  ? 


134  ORAL  LESSONS  IN  NUMBER. 

3.  1  time    6?  6  in     6? 

2  times  6?  6  in  12? 

3  times  6?  6  in  18? 

4  times  6?  6  in  24? 

5  times  6?  6  in  30? 

6  times  6?  6  in  36? 

7  times  6?  6  in  42? 

8  times  6?  6  in  48? 

9  times  6?  6  in  54? 
10  times  6?  6  in  60? 

How  many  are  3  times  6?  5  times  6?  7  times  6? 
9  times  6?  4  times  6?  6  times  6?  8  times  6?  10 
times  6? 

How  many  times  6  in  12  ?  6  in  24?  6  in  36?  6  in 
48?  6  in  60?  6  in  18?  6  in  42?  6  in  30?  6  in  54? 


4.  A  beetle  has  6  legs :  how  many  legs  have  3 
beetles?  5  beetles? 

An  orchard  contains  6  rows  of  peach  trees,  and  there 
are  6  trees  in  each  row :  how  many  peach  trees  in  the 
orchard  ? 

A  school-room  has  7  rows  of  desks,  with  6  desks  in 
each  row  :  how  many  desks  in  the  room  ? 

Jane  wrote  10  words,  and  each  word  contained  6  let- 
ters :  how  many  letters  did  she  write  ? 

John  made  8  columns  of  figures  on  his  slate,  and 
each  column  had  6  figures :  how  many  figures  did  he 
make  ? 

How  much  will  6  slates  cost  at  6  cents  apiece?  8 
slates  at  6  cents  apiece? 

How  many  slates,  at  6  cents  each,  can  be  bought  for 
30  cents?  For  60  cents? 

What  will  7  heads  of  cabbages  cost  at  6  cents  a 
head? 


THIRD-TEAR  COUI. 

How  many  melons,  at  6  cents  each,  ca 
for  42  cents?    For  54  cents? 

n  be  bought 

5.       6=   IX     ,  or6x     .'.      6  —  6  = 
12  =   2  X     ,  or  6  X     -  -     12  —  6  = 
18=   3  X     ?or6x     .'.     18  —  6  = 
24  =   4  X     ?  or  6  X     -  -     24  —  6  = 
30=  5  X     ,or6x     .'.    30  —  6  = 
36=   6X                       /.    36  —  6  = 
.=    7X     .  or6x     A     42-6  = 
48=   8X     ,or6x     .'.    48  —  6  = 

6-    1  = 

12-    2  = 
18—    3  = 
24—    4  = 
30—   5  = 

42-    7  = 

^-    -  = 

M=   9X     ,or6x     /.    54  —  6  = 

54—9  = 

60  =  10X     ,or6x     -•     60  —  6=    ; 

60  --10  = 

Xom—  See 

Leaaon  IH.  page  125,  note. 

SLATE    AND   BOARD   EXERCISES. 

1X6  = 

6  —  6  = 

3X6  = 

18-^6  = 

2X6  = 

12  —  6  = 

1X6  = 

6-6  = 

3x6  = 

18—6  = 

4X6  = 

24  —  6  = 

4X6  = 

24  —  6  = 

2X6^ 

12—6  = 

5X6  = 

30  —  6  = 

5X6  = 

30—6  = 

6X6  = 

36  —  6  = 

7X6  = 

42-6  = 

7X6  = 

42  —  6  = 

9X6  = 

54-6  = 

8X6  = 

48  —  6  = 

8X6  = 

48  —  6  = 

9X6  = 

54-6  = 

6x6  = 

36—6  = 

10X6  = 

60—6  = 

10x6  = 

60  —  6  = 

a 

b          r          d         e         f 

<7            h 

6 

66666 

6         6 

Multiply     3 

5469 

7         8 

a          b          c  d          f          f          f)        h 

Dicidc  6)18    6)12    6)30    6)24    6)36    6)54    6)42    6)48 


136 


ORAL  LESSORS  IN  NUMBER. 


12 


a                         A 

18\ 

\ 
\ 

48—         _x  * 

/' 

/            \ 
& 

T±O-—            —    t 

5/ 
3 

3_ 
\ 
\2 
6 

LESSON  VII. 

1.  How  many  balls  are  : 

Two      7  balls? 

2  times  7  balls? 

Three   7  balls? 

3  times  7  balls? 

Four     7  balls? 

4  times  7  balls? 

Five     7  balls? 

5  times  7  balls? 

Six       7  balls? 

6  times  7  balls? 

Seven   7  balls? 

7  times  7  balls? 

Eight    7  balls? 
Nine     7  balls? 

8  times  7  balls? 
9  times  7  balls? 

Ten       7  balls? 

10  times  7  balls? 

NOTE,  —  See  Lesson  II,  page 

120,  note. 

2.  How  many  are : 

3  times  7  marbles? 

5  times  7  marbles? 

7  times  7  pencils? 

4  times  7  pencils? 

6  times  7  soldiers? 

8  times  7  soldiers? 
2  times  7  figures? 

9  times  7  dollars? 
10  times  7  dollars  ? 


How  many  times  : 
7  marbles  in  21  marbles? 
7  marbles  in  35  marbles? 
7  pencils  in  49  pencils? 
7  pencils  in  28  pencils? 
7  soldiers  in  42  soldiers? 
7  soldiers  in  56  soldiers? 
7  figures  in  14  figures? 
7  dollars  in  63  dollars  ? 
7  dollars  in  70  dollars? 


THIRD- YEAR  COURSE.  137 

3.         1  time    7?  7  in    7? 

2  times  7?  7  in  14? 

3  times  7?  7  in  21? 

4  times  7?  7  in  28? 

5  times  7?  7  in  35? 

6  times  7?  7  in  42? 

7  times  7?  7  in  49? 

8  times  7?  7  in  56? 

9  times  7?  7  in  63? 
10  times  7?  7  in  70? 

How  many  are  3  times  7?  5  times  7?  Twice  7?  4 
times  7  ?  8  times  7  ?  6  times  7  ?  9  times  7  ?  7  times 
7?  10  times  7? 

How  many  times  7  in  21  ?  7  in  35?  7  in  14?  7  in 
28?  7  in  56?  7  in  42?  7  in  63?  7  in  49?  7  in 
70? 


4,  There  are  7  days  in  a  week :  how  many  days  in  4 
weeks?  In  8  weeks?  6  weeks? 

If  there  are  7  boards  in  a  length  of  fence,  how  many 
boards  will  make  5  lengths  ?  9  lengths  ? 

There  are  7  hills  of  tomatoes  in  a  row:  how  many 
hills  in  7  rows?  In  10  rows? 

If  a  slate  cost  7  cents,  how  many  cents  will  6  slates 
cost?  9  slates? 

How  many  slates,  at  7  cents  each,  can  be  bought  for 
35  cents?  49  cents?  63  cents? 

How  much  will  6  melons  cost  at  7  cents  apiece?  10 
melons  at  7  cents  apiece? 

How  many  yards  of  muslin,  at  7  cents  a  yard,  can  be 
bought  for  42  cents?  For  70  cents? 

How  many  pounds  of  sugar,  at  7  cents  a  pound,  can 
be  bought  for  63  cents?  For  56  cents? 

0&L.-12. 


138 

ORAL 

LESSONS  IN 

NUMBER. 

5,     7  = 

IX 

,  or 

7 

X     /. 

7 

_:_ 

7  —     • 

; 

7-^ 

1 

= 

14  = 

2X 

i  or 

7 

X     /. 

14 



n  

14-- 

2 

rr= 

o-j   

3X 

,  or 

7 

X     /. 

21 



7=    \ 

21-5- 

3 

= 

28  = 

4X 

,  or 

7 

x    .-. 

28 

— 

7  =     ; 

28-5- 

4 

= 

35  = 

5X 

,  or 

7 

x    .-. 

35 



7  =    ; 

35-5- 

5 

=. 

42  = 

6X 

i  or 

7 

x   .-. 

42 



7  —     • 

'                     7 

42-5- 

6 

= 

49  = 

7X 

/t 

49 

— 

7  = 

56  = 

8X 

,  or 

7 

x   .-. 

56 

— 

7—     • 
) 

56-- 

8 

— 

63  = 

9X 

,  or 

7 

x   .-. 

63 

— 

7=    ; 

63- 

9 

= 

70  =  10  X 

,  or 

7 

x   /. 

70 

7  —     • 
j 

70- 

10 

SLATE 

AND   BOARD    EXERCISES. 

IX 

7 

7-7  = 

2 

X 

7  = 

14  — 

7  = 

2X 

7 

14  —  7  = 

4 

X 

7  = 

28-^- 

7  = 

3X 

7 

21—7  = 

1 

x 

7  = 

7  — 

7  = 

4X 

7 

28  —  7  = 

3 

X 

7  — 

21  — 

7  = 

5X 

7 

35—7  = 

5 

X 

7  = 

35  — 

7  = 

6X 

7 

42—7  = 

7 

X 

J7  

49  —  7  = 

7X 

7 

= 

49- 

-7 

=3 

6 

X 

7  = 

42  — 

7: 

8X 

7 

56.—  7  = 

8 

X 

fr  

56  — 

fr  

9X 

7 

= 

63- 

-7 

= 

10 

X 

7  — 

70  — 

7  = 

10  X 

7 

70-7  = 

9 

X 

7  = 

63  — 

7  — 

a 

b 

c 

d 

*     f 

g 

h 

7 

7 

7 

7 

7         7 

7 

7 

Multiply 

3. 

4 

2 

5 

7       _9 

6 

8 

abcdefgh 
Divide    7)21    7)28    7)14    7)35    7)49    7)63    7)42    7)56 


THIRD- YEAR  COURSE. 


139 


14 


8  — 


21 


56 7 28 


42 


LESSON  VIII. 

1.  How  many  balls  are : 

Two     8  balls?  2  times 

Three  8  balls?  3  times 

Four    8  balls?  4  times 

Five     8  balls?  5  times 

Six       8  balls?  6  times 

Seven  8  balls?  7  times 

Eight  8  balls?  8  times 

Nine    8  balls?  9  times 

Ten      8  balls?  10  times 

NOTE. — See  Lesson  II,  page  120,  note. 


8  balls? 
8  balls? 
8  balls? 
8  balls? 
8  balls? 
8  balls? 
8  balls? 
8  balls? 
8  balls? 


2.  How  many  are: 

3  times  8  quarts  ? 

5  times  8  quarts  ? 
2  times  8  yards  ? 

4  times  8  yards  ? 

6  times  8  pounds? 

8  times  8  pounds  ? 
10  times  8  pounds  ? 

7  times  8  dimes  ? 

9  times  8- dimes? 


How  many  times : 
8  quarts  in  24  quarts  ? 
8  quarts  in  40  quarts  ? 
8  yards  in  16  yards  ? 
8  yards  in  32  yards  ? 
8  pounds  in  48  pounds? 
8  pounds  in  64  pounds  ? 
8  pounds  in  80  pounds? 
8  dimes  in  56  dimes? 
8  dimes  in  72  dimes? 


140  ORAL  LESSONS  IN  NUMBER 

3.  1  time    8?  8  in    8? 

2  times  8?  8  in  16? 

3  times  8?  8  in  24? 

4  times  8?  8  in  32? 

5  times  8?  8  in  40? 

6  times  8?  8  in  48? 

7  times  8?  8  in  56? 

8  times  8?  8  in  64? 

9  times  8?  8  in  72? 
10  times  8?  8  in  80? 

How  many  are  3  times  8?  7  times  8?  5  times  8? 
9  times  8?  4  times  8?  8  times  8?  2  times  8?  6 
times  8?  10  times  8? 

How  many  times  8  in  24?  8  in  56?  8  in  40?  8  in 
72?  8  in  32?  8  in  64?  8  in  16?  8  in  48?  8  in  80? 


4.  There  are  8  quarts  in  a  peck  :  how  many  quarts 
in  5  pecks?  In  7  pecks?  In  10  pecks? 

If  there  be  8  panes  of  glass  in  1  window,  how  many 
panes  of  glass  in  4  windows?  8  windows? 

A  school-room  has  6  rows  of  desks,  with  8  desks  in 
each  row :  how  many  desks  in  the  school-room  ? 

Helen  has  written  9  columns  of  figures  on  her  slate, 
with  8  figures  in  each  column :  how  many  figures  has 
she  written? 

A  railroad  car  has  8  wheels :  how  many  wheels  in  a 
train  of  6  cars? 

How  many  pecks  in  40  quarts?     In  80  quarts? 

If  a  man  work  8  hours  a  day,  how  many  hours  will 
he  work  in  6  days? 

How  many  melons,  at  8  cents  each,  can  be  bought 
for  56  cents?  For  64  cents? 

How  many  pounds  of  sugar,  at  8  cents  a  pound,  can 
be  bought  for  40  cents?  For  72  cents? 


THIRD- YEAR  COURSE. 


141 


5. 


8 

^= 

1 

X 

,  or 

8X     /. 

8  —  8=    ; 

8-4- 

1: 

16 

= 

2 

X 

,  or 

8X     .'. 

16  —  8=    ; 

16  — 

2  = 

24 

== 

3 

X 

,  or 

8X     /. 

24  —  8=    ; 

24  — 

3  = 

32 

sss 

4 

X 

,  or 

8X     .'. 

32- 

8. 
—      J 

32  — 

4: 

40 

= 

5 

X 

,  or 

8X     /. 

40- 

—      J 

40- 

5  = 

48 

=    6X 

,  or 

8X     .'. 

48- 

-8=     ; 

48- 

6  = 

56 

= 

7 

X 

>  or 

8X     .'. 

56- 

-8=    ;  56—   7  = 

64 

= 

8 

X 

t"m 

64- 

-8  = 

72 

= 

9 

X 

i  or 

8X     /. 

72- 

-8=    ;   72—    9  = 

80 

10 

X 

,  or 

8X     /• 

80- 

-8=  ; 

80- 

10  = 

SLATE 

AND   BOARD   EXERCISES. 

1 

X 

8  = 

8 

= 

3 

X8  = 

24 

4-8 

— 

2 

X 

8: 

16 

—  8  = 

1 

X  8  = 

8 

-5-8 

= 

3 

X 

8: 

24 

—  8  = 

4 

X8  = 

32 

-5-8 

== 

4 

X 

8: 

32 

—  8  = 

2 

X8  = 

16 

-4-8 

= 

5 

X 

8: 

40 

n  

5 

X  8  = 

40 

—  8 

= 

6 

X 

8  = 

48 

—  8  = 

8 

X8  = 

64 

—  8 

= 

7 

X 

8: 

56 

—  8  = 

6 

X  8  = 

48 

—  8 

= 

8 

X 

8: 

64 

—  8  = 

9 

X  8  = 

72 

—  8 

= 

9 

X 

8  = 

72 

o  

7 

X8  = 

56 

—  8 

= 

10 

X 

8: 

80 

-4-8  = 

10 

X8  = 

80 

—  8 

a 

b 

c 

d 

e         f 

9 

h 

8 

8 

8 

8 

8         8 

8 

8 

Mi 

oly 

3 

2 

4 

5 

1         9 

6 

8 

abode  f         9         h 

Divide   8)24    8)16     8)32    8)40    8)56    8)72    8)48    8)64 


142 


ORAL  LESSONS  IN  NUMBER. 


16 


\ 

8< 

. 

X                                                                                   \ 

/ 

x,40 

X 

/ 
/ 
/ 
>                           rtf> 

•>•'' 

N                                             / 

\                                                        / 
\                                                   / 

N7                    72'' 

>  s       -    32 
\ 

N56 

LESSON  IX. 


1.  How  many  balls  are: 
Two     9  balls? 
Three  9  balls? 
Four    9  balls? 

9  balls? 

9  balls? 

9  balls? 

9 


Five 

Six 

Seven 

Eight  9  balls? 

Nine    9  balls? 

Ten      9  balls? 


2  times 

3  times 

4  times 

5  times 

6  times 

7  times 

8  times 

9  times 
10  times 


9  balls? 
9  balls? 
9  balls? 
9  balls? 
9  balls? 
9  balls? 
9  balls? 
9  balls? 
9  balls? 


NOTE. — See  Lesson  II,  page  120,  note. 


2.  How  many  are: 

3  times  9  miles? 

7  times  9  miles? 

4  times  9  inches? 
6  times  9  inches? 
2  times  9  pints? 

5  times  9  pints? 

8  times  9  rings? 
10  times  9  rings? 

9  times  9  rings? 


How  many  times: 
9  miles  in  27  miles? 
9  miles  in  63  miles? 
9  inches  in  36  inches? 
9  inches  in  54  inches? 
9  pints  in  18  pints? 
9  pints  in  54  pints? 
9  rings  in  72  rings? 
9  rings  in  90  rings? 
9  rings  in  81  rings? 


THIRD- YEAR  COURSE.  143 

3,  1  time    9?  9  in     9? 

2  times  9?  9  in  18? 

3  times  9?  9  in  27? 

4  times  9?  9  in  36? 

5  times  9?  9  in  45? 

6  times  9?  9  in  54? 

7  times  9?  9  in  63? 

8  times  9?  9  in  72? 

9  times  9?  9  in  81? 
10  times  9?  9  in  90? 

How  many  are  3  times  9?  7  times  9?  5  times  9? 
9  times  9?  4  times  9?  2  times  9?  8  times  9?  6 
times  9?  10  times  9? 

How  many  times  9  in  27  ?  9  in  63  ?  9  in  45  ?  9  in 
81?  9  in  36?  9  in  18?  9  in  72?  9  in  54?  9  in 
90? 


4,  An  orchard  has  9  rows  of  trees,  and  9  trees  in 
each  row :  how  many  trees  in  the  orchard  ? 

Five  of  the  rows  of  trees  in  the  orchard  are  apple 
trees,  and  4  of  them  are  peach  trees :  how  many  apple 
trees  in  the  orchard?  How  many  peach  trees? 

If  a  man  work  9  hours  a  day,  how  many  hours  will 
he  work  in  6  days?  In  10  days? 

How  many  dollars  will  7  sheep  cost  at  9  dollars  a 
head?  8  sheep?  5  sheep? 

At  9  dollars  a  head,  how  many  sheep  can  be  bought 
for  45  dollars?  For  63  dollars?  For  36  dollars? 

How  many  cents  will  6  pounds  of  sugar  cost  at  9 
cents  a  pound?  8  pounds? 

At  9  cents  a  pound,  how  .many  pounds  of  sugar  can 
be  bought  for  54  cents?  For  72  cents? 

How  many  bunches  will  63  radishes  make,  if  9 
radishes  be  put  in  each  bunch? 


144 


OEAL  LESSONS  IN  NUMBER. 


9  = 

1 

X 

,  or  9 

X     .'.      9 

-£- 

9=    ; 

9-   1 

18  = 

2 

X 

,  or  9 

X     /.     18 

-H 

9=    ; 

18—2 

27  = 

3 

X 

,  or  9 

X     /.     27 

-£- 

9=    ; 

27—3 

36  = 

4 

X 

,  or  9 

X     /.     36 

-f- 

9=    ; 

36—4 

45  = 

5 

X 

,  or  9 

X     /.    45 

-f- 

9=    ; 

45—5 

54  = 

6 

X 

,  or  9 

X     /.    54 

•4- 

9=    ; 

54—6 

63  = 

7 

X 

,  or  9 

X     /.    63 

-JH 

9=    ; 

63—7 

72  = 

8 

X 

,  or  9 

X     /.     72 

H- 

9=    ; 

72—8 

81  = 

9 

X 

/.     81 

-4- 

9  = 

90  = 

10 

X 

,  or  9 

X    /.    90 

9=    ; 

90  --10 

SLATE   AND   BOARD   EXERCISES. 

IX 

9 

= 

9  —  9 

1 

2 

X 

9  = 

18  —  9: 

2X 

9 

= 

18- 

-9 

ass 

4 

X 

Q  

36- 

-9  = 

3X 

9 

= 

27- 

-9 

— 

6 

X 

9  = 

54- 

-9  = 

4X 

9 

= 

36- 

-9 

±= 

1 

X 

9  = 

9- 

-9  = 

5X 

9 

= 

45- 

-9 

= 

3 

X 

9  = 

27- 

-9^ 

6X 

9 

= 

54- 

-9 

= 

5 

X 

9  = 

45- 

-9  = 

7X 

9 

— 

63- 

-9 

= 

8 

X 

9  = 

72- 

-9  = 

8X 

9 

= 

72- 

-9 

= 

10 

X 

9  = 

90- 

-9  = 

9X 

9 

= 

81- 

-9 

— 

7 

X 

9  = 

63- 

-9  = 

10  X 

9 

= 

90- 

-9 

= 

9 

X 

9  = 

81- 

-9  = 

NOTE. — The  successive  products  of  9  by  the  other  digital 
numbers  are  9,  18,  27,  36,  45,  54,  63,  72,  81,  and  90.  It  is  an 
interesting  fact  that  the  sum  of  the  digits  that  express  each  of 
these  products  is  9.  It  is  also  seen  that,  in  the  successive  pro- 
ducts, the  ten  figure  increases  by  one,  and  the  unit  figure  de- 
creases by  one. 


Multiply 


f  9 
9  9 
9  6 


THIRD- YEAR  COURSE. 


145 


abed  e         f          g          h 

Divide  9)18    9)36    9)63    9)27     9)45     9)81     9)54     9)72 


NOTE. — Additional  blackboard  exercises  in  multiplying  num- 
bers, written  in  horizontal  lines,  may  be  easily  provided  by 
writing  a  horizontal  line  of  figures  on  the  board,  and  having  the 
number  expressed  by  each  figure,  beginning  at  the  right,  mul- 
tiplied by  a  designated  digital  number,  as  4,  7,  9,  etc.  The 
division  of  the  successive  products  by  each  of  its  factors  will 
afford  excellent  practice  in  division. 


\ 


\ 


J/ 

8 9 4 


/ 1  \ 


18 


27 


72 9 


81' 


N63 


54 


LESSON  X. 


1.  How 

Two 

Three 

Four 

Five 

Six 

Seven 

Eight 

Nine 

Ten 


many  balls  are 
10  balls? 
10  balls? 
10  balls? 
10  balls? 
10  balls? 
10  balls? 
10  balls? 
10  balls? 
10  balls? 


2  times 

3  times 

4  times 

5  times 


6  times 

7  times 

8  times 

9  times 
10  times 


10  balls? 
10  balls? 
10  balls? 
10  balls? 
10  balls? 
10  balls? 
10  balls? 
10  balls? 
10  balls? 


NOTE.— See  Lesson  II,  page  120,  note. 

O.  L.-13. 


146 


ORAL  LESSONS  IN  NUMBER. 


2.  How  many  are  : 

How  many  times  : 

3  times  10  dimes  ? 

10  dimes  in  30  dimes  ? 

5  times  10  dimes? 

10  dimes  in  50  dimes  ? 

7  times  10  dollars  ? 

10  dollars  in  70  dollars  ? 

4  times  10  dollars? 

10  dollars  in  40  dollars  ? 

2  times  10  fingers? 
6  times  10  fingers? 
8  times  10  cents  ? 

10  fingers  in  20  fingers? 
10  fingers  in  60  fingers? 
10  cents  in  80  cents  ? 

9  times  10  cents? 

10  cents  in  90  cents  ? 

10  times  10  cents  ? 

10  cents  in  100  cents? 

3.          1  time    10? 

10  in     10? 

2  times  10? 

10  in     20? 

3  times  10? 

10  in     30? 

4  times  10? 

10  in    40? 

5  times  10? 

10  in    50? 

6  times  10? 

10  in     60? 

7  times  10? 

10  in     70? 

8  times  10? 

10  in     80? 

9  times  10? 

10  in     90? 

10  times  10? 

10  in  100? 

4  times 

5  times 


How  many  are  3  times  10?  7  times  10? 
10?  2  times  10?  6  times  10?  8  times  10? 
10?  9  times  10?  10  times  10? 

How  many  times  10  in  30?  10  in  70?  10  in  40? 
10  in  20?  10  in  60?  10  in  80?  10  in  50?  10  in  90? 
10  in  100? 


4.  There  are  10  cents  in  a  dime :  how  many  cents  in 
5  dimes?  3  dimes?  10  dimes?  « 

How  many  dimes  in  50  cents?  In  30  cents?  In  60 
cents?  In  100  cents? 

There  are  10  dimes  in  a  dollar :  how  many  dimes  in 
4  dollars?  In  6  dollars?  In  8  dollars? 


THIRD- YEAR   COURSE. 


147 


How  many  dollars  in  40  dimes?  In  60  dimes?  In 
80  dimes?  In  100  dimes? 

There  are  10  postage  stamps  in  a  row,  and  10  rows  in 
a  sheet :  how  many  postage  stamps  in  a  sheet  ? 

How  many  postage  stamps  in  5  rows  or  one  half  of  a 
sheet?  In  7  rows?  In  9  rows? 

What  is  the  cost  of  6  yards  of  calico,  at  10  cents  a 
yard?  7  yards?  10  yards? 

How  many  pounds  of  cheese,  at  10  cents  a  pound, 
can  be  bought  for  60  cents?  For  70  cents?  100  cents? 


5. 

10  = 

1 

X 

,  or 

10 

x   .-.   10- 

5-10  = 

;    10-1 

20  = 

2 

X 

i  or 

10 

X     /.     20- 

-10  = 

;    20  —  2 

30  = 

3 

X 

,  or 

10 

X     /.     30- 

-10  = 

;    30  —  3 

40  = 

4 

X 

i  or 

10 

X     .'.     40- 

-10  = 

;    40  —  4 

50  = 

5 

X 

,  or 

10 

X     /.    50- 

-10  = 

;    50  —  5 

60  = 

6 

X 

,  or 

10 

X     /.     60- 

-10  = 

;    60  —  6 

70  = 

7 

X 

i  or 

10 

X     .'.     70- 

-10  = 

;    70-7 

80  = 

8 

X 

,  or 

10 

X     ..%    80- 

-10  = 

;    80  —  8 

90  = 

9 

X 

,  or 

10 

X     .'.    90- 

-10  = 

;   90  —  9 

100  —  10 

X 

.-.  100-5-10  = 

SLATE    AND 

BOARD    EXERCISES. 

1 

xio= 

10  — 

10  = 

3X10  = 

30- 

-10 

2 

xio  = 

20- 

10  = 

6X10  = 

60- 

-10 

3 

xio= 

30- 

10  = 

2X10  = 

20- 

-10 

4 

xio= 

40- 

10  = 

5X10  = 

50- 

-10 

5 

xio  = 

50- 

10  = 

1X10  = 

10- 

-10 

6 

xio  = 

60- 

10  = 

4X10  = 

40- 

-10 

•  7 

xio= 

70- 

10  = 

8X10  = 

80- 

-10 

8 

xio  = 

80- 

10  = 

10X10  = 

100- 

-10 

9 

xio  = 

90- 

10= 

7X10  = 

70- 

-10 

10 

X  10  = 

100- 

10=            9X10  = 

90- 

-10 

148 


ORAL  LESSONS  IN  NUMBER 


a    b     c    d    u     / 

10    10    10    10    10    10 

Multiply    247359 


a 


9 

10 

6 


h 

10 

8 


b          c          d          e  f          g          h 

Divide  10)20  10)40  10)70  10)30  10)50  10)90  10)60  10)80 


,5 


8 10 4 


X7 


30, 


20 


80 10 40 


90' 


,50 


N70 


60 


LESSON  XI. 

1.  There  are  only  forty-five  primary  combinations  in 
multiplication,  and  these  are  formed  by  multiplying 
each  digital  number  (10  not  included)  by  itself  and  by 
each  of  the  higher  digital  numbers,  as  follows: 

a          b  c  d          e          f          g         h          i 

1X1  2X2  3X3  4X4  5x5  6x6  7x7  8x8  9x9 

2X1  3X2  4X3  5X4  6x5  7x6'  8x7  9x8 

3X1  4x2  5x3  6x4  7x5  8x6  9X7 

4X1  5X2  6X3  7X4  8x5  9x6 

5X1  6X2  7X3  8x4  9x5 

6X1  7X2  8X3  9X4 

7X1  8X2  9X3 

8X1  9X2 

9X1 


THIRD- YEAR  COURSE. 


149 


NOTE. — These  forty-five  combinations  may  be  written  on  the 
board,  and  the  pupils  drilled  until  they  give  the  products  in- 
stantly. 


2.  These  primary  combinations  in  multiplication, 
and  the  corresponding  inverse  processes  in  division, 
may  be  recited  as  follows: 


2X1  or  1X2  = 
3X1  or  1X3  = 
4x1  or  1x4  = 
5X1  or  1x5  = 
6X1  or  1x6  = 
7X1  or  1X7  = 
8X1  or  1X8  = 
9X1  or  1X9  = 


1  =  1 


2X2  =  4 
3X2  or  2X3  = 
4x2  or  2X4  = 
5X2  or  2x5  = 
6  X  2  or  2  X  6  = 
7X2  or  2X7  = 
8X2  or  2X8  = 
9X2  or  2X9  = 


3X3  = 

4X3  or  3X4  = 
5x3  or  3x5  = 
6x3  or  3X6  = 
7X3  or  3X7  = 
8x3  or  3X8  = 
9x3  or  3X9  = 


4--2=2 

/> 

-2  = 

/» 

-3 

O  ~ 

-2  = 

8- 

-4 

10- 

—  2  = 

10- 

-5 

12- 

-2  = 

12- 

-6 

14- 

-2  = 

14- 

-7 

16- 

-2  = 

16- 

-8 

18- 

-2=       ; 

18- 

-9 

9- 

-3  =  3 

12- 

-3  = 

;  12- 

-4 

15- 

-3  = 

;  15- 

-5 

18- 

0  

;  is- 

-6 

21- 

-3  = 

;  21- 

-7 

24- 

o  

;   24- 

-8 

27^-3  = 

;  27- 

-9 

150  ORAL  LESSONS  IN  NUMBER. 

4X4  =  16  .-.  16--4  =  4 

5X4  or  4X5=  .-.  20--4  =    ;  20-=-5  = 

6X4  or  4X6  =  /.  24-r-4  =     :  24-^-6  = 

7X4  or  4X7=  /.  28-5-4  =    ;  28-5-7  = 

8X4  or  4X8  =  /.  32-f-4  =    ;  32-5-8  = 

9X4  or  4X9=  .-.  36-v-4  =    ;  36-5-9  = 

5x5  =  25  •-.  25-5-5  =  5 

6X5  or  5X6=  r.  30 -s- 5=    ;  30-5-6  = 

7X5  or  5x7=  /.  35^5=    ;  35-5-7  = 

8X5  or  5X8=  /.  40-5-5=    ;  40-*- 8  = 

9X5  or  5X9=  .-.  45-^-5=    ;  45-^-9  = 

6X6  =  36  /.  36^6  =  6 

7X6  or  6X7=  /.  42^6=    ;  42-5-7  = 

8X6  or  6X8=  .%  48-5-6=    ;  48n-8  = 

9X6  or  6X9=  .%  54 -s- 6=    ;  54-5-9  = 

7x7  —  49  .-.  49^7  =  7 

8x7  or  7X8=  /.  56^-7=    ;  56-=-8  = 

9X7  or  7X9=  /.  63--7—    ;  63-5-9  = 

8x8=64  /.  64 -4-8  =  8 

9X8  or  8X9=  .-.  72-^-8=    ;  72^9  = 

9x9  =  81  /.  81-5-9  =  9 


LESSON  XII. 

Parts  of  Numbers. 

The  previous  lessons  have  made  the  pupils  familiar 
with  the  process  of  separating  a  number  into  equal 
parts.  The  step  now  to  be  taken  is  to  teach  pupils  to 
find  directly  one  of  the  equal  parts  of  a  number,  and 
this  involves  the  developing  of  the  idea  of  one  half,  one 
third,  one  fourth,  and  other  fractional  parts  of  a  unit. 


THIRD-YEAR   COURSE.  151 

1.  Take  an  apple  and  cut  it  into  two  equal  pieces,  and 
teach  that  one  of  these  pieces  is  one  half  of  the  apple. 
Write  the  fraction  %  on  the  board. 

Cut  the  apple  into  four  equal  pieces  by  cutting  each 
half  into  two  equal  pieces,  and  teach  that  each  one  of 
these  pieces  is  one  fourth  of  the  apple ;  that  two  pieces 
are  two  fourths ;  three  pieces,  three  fourths,  etc.  Write 
the  fraction  J  on  the  board. 

How  many  halves  in  an  apple  ?  How  many  fourths  ? 
How  many  fourths  in  one  half? 

Cut  the  apple  into  eight  equal  pieces  by  cutting  each 
fourth  into  two  equal  pieces,  and  teach  that  each  one 
of  these  eight  pieces  is  one  eighth  of  the  apple;  that 
two  pieces  are  two  eighths;  three  pieces,  three  eighths,  etc. 
Write  the  fraction  -J  on  the  board. 

How  many  eighths  in  an  apple?  How  many  eighths 
in  one  half  of  it?  In  one  fourth? 

Draw  a  line  and  a  circle  on  the  board,  and  divide  each 
into  halves,  fourths,  and  eighths,  and  have  the  pupils 
name  the  parts  as  each  is  divided. 

Next  take  an  apple  and  divide  it  into  three  equal 
pieces,  and  teach  that  each  piece  is  one  third  of  the 
apple.  Write  the  fraction  -J  on  the  board. 

In  like  manner  teach  one  sixth,  one  ninth,  one  fifth,  one 
tenth,  etc. 


2.  How  much  is  £  of  2  blocks  ?  £  of  4  blocks  ?  |  of 
8  blocks  ?  i  of  10  blocks  ?  |  of  12  blocks  ? 

NOTE.— Take  the  number  of  blocks  named  in  the  question, 
and  separate  them  into  two  equal  groups. 

How  much  is  i  of  4  blocks?  \  of  8  blocks?  \  of  12 
blocks?  i  of  16  blocks?  \  of  20  blocks? 

How  much  is  i  of  3  blocks?  i  of  9  blocks?  £  of  12 
blocks?  of  18  blocks?  of  24  blocks? 


152  ORAL  LESSONS  IN  NUMBER. 

How  much  is  i  of  12  balls?  £  of  18  balls?  £  of  30 
balls?  i  of  24  balls?  1  of  36  balls?  I  of  48  balls? 

How  much  is  i  of  10?  4- of  20?  £  of  30?  |of40? 
|  of  35?  £  of  45?  J.  of  50? 

NOTE. — The  above  exercises  should  be  multiplied  until  the 
pupils  are  familiar  with  the  process  of  taking  £,  £,  £,  |,  £,  etc., 
of  a  number. 


3.  The  following  exercises  may  be  written  on  the 
blackboard,  and  used  for  oral  drills  in  the  analysis  of 
numbers  into  equal  parts,  and  then  finding  one  of  the 
equal  parts. 

4  — two  2's  /.  i  of    4  =  2 

6=       2's   or      3's  .'.  £  of    6=    ;   i  of    6  = 

8—       2's   or      4's  .%  J  of    8=    ;  \  of    8  — 

10=        2's    or      5's  /.  £of  10  =     ;   £  of  10  = 

12—       2's    or       6's  /.  J  of  12—    ;   £of  12  = 

14—  2's    or       7's  /.  f  of  14—    ;   £  of  14  = 
16—       2's    or       8's  A  \  of  16—    ;   ^  of  16  = 
18=        2's    or       9's  /.  \  of  18—    ;   £  of  18  — 

9  —  three  3's  .'.  \  of    9  —  3 

12—       3's   or       4's  /.  \  of  12  —    ;|ofl2  — 

15—  3's    or      5's  .'.  \  of  15  —     ;   \  of  15  = 
18—        3's    or       6's  .-.  $  of  18-     ;   ^  of  18  = 
21  —       3's    or      7's  /.  \  of  21  —    ;   \  of  21  = 
24—       3's   or       8's  /.  |  of  24—     ;   |  of  24  — 
27—       3's    or      9's  /.  £  of  27—    ;   iof27  = 

16  — four  4's  .'.  J  of  16  —  4 

20—       4's    or      5's  .'.  |  of  20—    ;   ^  of  20  — 

24—       4's   or      6's  /.  Jof24=      ;   J  of  24  = 

28  —       4's    or       7's  /.  }  of  28  —    ;   ±  of  28  = 

32  —       4's    or      8's  /.  |  of  32  —    ;   \  of  32  — 

36—       4's   or      9's  /.  |of36—    ;   1  of  36  — 


THIRD- YEAR  COURSE.  153 


25  =  five  5's 

.-.       iof  25  =5 

30  = 

5's    or      6s 

V,       iof  30=     ; 

iof  30  = 

35  = 

5's    or      7's 

/.      iof  35=     ; 

iof  35  = 

40  = 

5's   or      8's 

/.       iof  40=    : 

iof  40  = 

45  = 

5's    or      9's 

.-.   iof  45=  ; 

iof  45  = 

36  =  six  6's 

.'.      -J-  of  36  =  6 

42  = 

6's    or       7's 

.-.      iof  42=    ; 

iof  42  = 

48  = 

6's    or      8's 

.'.       iof  48=    ; 

iof  48  = 

54  = 

6's    or      9's 

i  of  54=    ; 

iof  54  = 

49  =  seven  7's 

.-.       i  of  49  =  7 

56  = 

7's    or      8's 

/.      iof  56=    ; 

i  of  56  = 

63  = 

7's   or      9's 

/.       i  of  63  =    ; 

i  of  63  = 

64  =  eight  8's 

/.      iof  64  =  8 

72  = 

8's    or       9's 

/.      iof  72=    ; 

iof  72  = 

81  =  nine  9's 

.-.      iof  81  =  9 

LESSON  XIII. 

Supplemental  Drills  in  Rapid  Combinations. 

The  foregoing  exercises  may  be  supplemented  by  oral 
and  blackboard  drills  in  the  rapid  combination  of  num- 
bers by  addition,  subtraction,  multiplication,  and  divi- 
sion. The  rapidity  secured  by  these  drills  has  won  for 
them  the  appellation  of  "Lightning  Combinations." 

1.  The  following  are  illustrations  of  oral  drills: 

1.  Take  5,  add  7,  subtract  4,  multiply  by  7,  add  4, 
divide  by   6,  subtract  3,  multiply  by  9,  add  5,  add  4, 
divide  by  8.     What  is  the  result? 

2.  Take  9,  multiply  by  6,  add  7,  subtract  5,  divide  by 
8,  add  2,  multiply  by  6,  add  9,  divide  by  9,  multiply 
by  5,  add  7,  divide  by  7,  multiply  by  8.     Result? 

3.  Take  65,  subtract  9,  divide  by  8,  multiply  by  6, 
add  9,  subtract  6,  divide  by  9,  add  4,  multiply  by  8, 
add  9,  divide  by  9,  divide  by  3.     Result? 


154  ORAL  LESSONS  IN  NUMBER. 

4.  Take  15,  add  .12,  add  9,  subtract  4,  divide  by  8, 
add  4,  multiply  by  9,  subtract  8,  divide  by  8,  multiply 
by  7,  add  9,  add  7,  divide  by  8,  divide  by  3.  Result? 

NOTE.  —  At  first  the  class  may  be  permitted  to  give  orally  each 
result  ;  but,  as  soon  as  the  pupils  are  sufficiently  familiar  with 
the  process,  only  the  final  result  should  be  given.  The  teacher 
should  dictate  slowly  at  first,  increasing  in  rapidity  as  skill  is 
acquired  by  the  class. 

2.  The  drills  may  be  varied  by  writing  the  combina- 
tions on  tbe  blackboard,  and  pointing  successively  to 
those  to  be  made. 

The  following  exercises  are  illustrations  of  these 
blackboard  drills,  the  operations  being  performed  from 
left  to  right  in  order  : 

1.  19  +  8,  +9,  -4,  -4-8,  +4,  X9,  -8,  -8,  X  7,  +9, 
+  7,  -4-8,  -4-3  =  what? 

2.  8X9,   -12,   X7,    +7,    -4-7,    -f  3,    X  9,  —6,   -7, 


3.  56-7,   X9,  -5,  -4,  -7,   X  9,  +3,  -4-12,   X  6, 
-f  6,  -4-8,  -4-6  =  what? 

4.  42  +  6,    +9,  —3,  -4-9,   X  8,   -f  9,   -f  7,  -4-8,   X  10, 
+  1,  -4-9,  -4-  3  =  what? 

5.  6  +  5,  X7,  +4,  -4-9,  X7,  -5,  +6,  -4-8,  X  3,  +9, 
+  7,  -4-10,   X  12  =  what? 

6.  47  +  9,  -7,    X5,   +8,  —3,  -4-9,  X  12,  +3,  -7, 
X8,  -4-9,   X4  =  what? 

7.  100  —  25,  +6,  -4-9,  X6,  +10,  -4-8,  X  7,  +7,  -4-9, 
X7,  —1,  -4-  8  =  what? 

NOTE.  —  The  comma  (  ,  )  is  used  in  the  above  exercises  to  in- 
dicate that  the  operations  expressed  by  the  signs  +,  —  ,  X,  and 
H-  are  to  be  performed  in  the  order  in  which  they  occur  from  left 
to  right.  If  the  comma  were  not  thus  used,  the  operations  indi- 
cated by  X  and  -^  would  take  precedence  over  the  -f-  or  —  im- 
mediately preceding.  The  operations  in  the  first  exercise,  if 
written  without  the  comma,  would  be  performed  in  this  order: 
(5+7)  —  (4X7)  +  (4n-6)  —  (3X9)  +  5  +  (4-5-8).  The  sign  +  or  - 
would  include  all  that  lies  between  it  and  the  next  sign  of  +  or  —  . 


THIRD-YEAR   COURSE.  155 

Arithmeticians  are  not  agreed  respecting  the  order  in  which 
the  operations  indicated  by  the  signs  X  and  -=-  are  to  be  per- 
formed. 8X6-^4X3  may  equal  36  or  4,  since  the  operations 
may  be  performed  in  their  order  from  left  to  right,  or  8  X  6 
may  be  divided  by  4X3. 


SUPPLEMENTAL  BLACKBOARD  EXERCISES. 
ADDITION. 

The  exercises  in  adding  columns  given  in  the  ELE- 
MENTARY ARITHMETIC  (pages  18,  21,  23,  26,  28,  etc.), 
present  numerous  combinations  ;  and,  if  properly  used? 
will  afford  sufficient  slate  practice  for  third-year  pupils. 

1.  These   exercises   may    be   easily   re- 
produced and  multiplied  on  the  black-  1234 
board    by    writing   in    a   single    column 
the    additive    number    given,    and    then  4 
writing  successively  beneath  the  column  4 
each  of  the  lower  digits.  4 

Suppose,  for  illustration,  that   the  ad-  4 

ditive  number  is  4.    Write  a  column  of  4 

4's  on  the  blackboard  (as  at  the  right),  4 

and  under  the  column   (or  at  the  left)  4 

write  successively  1,  2,  and  3.  4 

The   use   of   1   as   the   initial   number  4 

will  give  as  results  5,  9,  13,  17,  21,  etc. ;  4 

the  use  of   2  as  the  initial  number  will  4 

give  6,  10,  14,  18,  22,  etc.;    and  the  use  4 

of  3  will  give  7,  11,  15,  19,  23,  etc.     In  4 

practice,    the    teacher    will    find    it    con-  4 

venient  to  erase  the  initial   figure  used,  4 

and  write  the  new  initial  figure  at  the  4 

foot  of  the  column.    The  columns  should  J_  _2_  _3  _4 
also     be    added    downwards,    beginning 
with  the  proper  initial  figure. 


4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

156  ORAL  LESSONS  IN  NUMBER. 

2.  These  exercises  are  all   included  in  the  following 
table : 

a  b  c  d  e  f          g          h  i 


1  2  3 
123 
1  2  3 
1  2  3 
123 
123 
123 
1  2  3 
123 
123 
1  2  3 
1  2  3 
1  2  3 
123 

i        2jL4A^L_LJL_?- 

In  adding  any  column  in  the  above  table,  the  initial 
numbers  of  the  columns  at  its  left  are  successively 
used;  thus,  the  column  of  6's  is  added  by  beginning 
successively  with  1,  2,  3,  4,  and  5. 

3.  Exercises  in  adding  columns  of  numbers  promiscu- 
ously arranged  may  be  readily  produced  on  the  black- 
board by  writing  a  column  of  the  lower  digital  num- 
bers and  the  number  reached,  arranged  promiscuously, 
and  then  writing  successively  at  the  foot  of  this 
column,  as  an  initial  number,  each  of  the  lower  digital 
numbers. 

These  promiscuous  exercises  are  shown  by  the  table 
on  the  opposite  page. 

In  adding  any  column  in  this  table,  begin  succes- 
sively with  the  initial  figures  of  the  columns  at  its 
left.  In  adding  the  sixth  (/)  column,  for  illustration, 
begin  successively  with  1,  2,  3,  4,  5,  and  6. 


THIRD-YEAR  COURSE.  157 

abcdefgh 

23356343 
12445787 
2  1  353668 
22224179 
12116586 
13223729 
23344414 
2  1  435429 
22456725 
13353389 
12341733 
23246249 
11232787 
22153659 
13424762 
12226479 
23433384 
12312269 


The  numbers  in  the  above  table  may  also  be  added 
horizontally  by  adding  the  different  lines  of  numbers, 
first  from  left  to  right,  and  then  from  right  to  left,  or 
inversely.  By  changing  the  initial  numbers,  the  exer- 
cises may  be  readily  multiplied. 

The  above  table  may  also  be  used  for  drills  in 
adding  the  digital  numbers  two  and  two.  The  first 
number  in  each  column  may  be  successively  added  to 
each  number  in  the  column,  progressing  upwards. 
Thus,  taking  the  fourth  (d)  column  for  illustration,  the 
number  5  may  be  successively  added  to  1,  3,  2,  2,  5, 
etc. 

The  first  number  at  the  top  of  each  column  may  also 
be  successively  added  to  each  number  in  the  column, 
progressing  downwards.  The  number  5  in  the  fourth 
(d)  column  may,  for  example,  be  successively  added  to 
4,  5,  2,  1,  etc. 


158  ORAL  LESSONS  IN  NUMBER. 

NOTES. — 1.  It  is  not  necessary  for  the  teacher  to  take  the  time  to 
copy  drill  exercises  on  the  blackboard.  Every  third-year  class  con- 
tains pupils  who  can  easily  be  trained  to  copy  plainly  and  neatly 
on  the  blackboard  all  the  drill  exercises  in  this  manual  and  also 
in  the  ELEMENTARY  ARITHMETIC,  and  this  copying  will  both 
please  and  benefit  the  pupils  who  do  it,  provided  the  work  be 
properly  distributed. 

2.  The  exercises  daily  produced  on  the  blackboard  for  class 
drill  are  fresher  and  otherwise  better  than  those  printed  on 
cards  or  charts.  It  is  also  questioned  whether  the  apparatus 
devised  by  different  persons  for  the  teaching  of  addition  is  not 
more  ingenious  than  useful.  Experience  shows  that  the  use  of 
such  devices  soon  becomes  very  mechanical,  and  that  the 
interest  of  the  pupils  lessens  as  the  sense  of  novelty  fades  out. 

II.  SUBTRACTION. 

The  first  and  second  year  exercises  have  given  pupils 
the  ability  to  subtract  instantly  any  digital  number 
from  the  sum  of  it  and  any  other  digital  number.  The 
third  year  exercises  in  subtraction  involve  the  taking 
of  the  digital  numbers  from  numbers  not  exceeding 
100,  and  the  only  difficulty  here  is  when  the  subtra- 
hend number  is  greater  than  the  number  denoted  by 
the  unit  figure  of  the  minuend,  as  in  subtracting  7 
from  31,  42,  53,  74,  65,  or  86. 

This  difficulty  is  easily  mastered  if  pupils  are  taught 
first  to  subtract  the  subtrahend  number  from  the  unit 
number  of  the  minuend  increased  by  ten,  and  then  take 
one  from  the  ten  number  of  the  minuend.  Thus,  31 
less  7  is  11  less  7  plus  30  less  10,  which  is  20,  plus  4, 
which  is  24.  42  less  7  is  12  less  7  plus  40  less  10, 
which  is  30,  plus  5,  which  is  35. 

Skill  in  these  processes  may  be  imparted  by  writing 
on  the  blackboard  series  of  numbers  with  the  same  unit 
figure,  and  then  subtracting  from  the  numbers  in  each 
series,  from  right  to  left,  the  several  digital  numbers 
successively,  as  in  the  table  at  top  of  opposite  page. 


THIRD-YEAR  COURSE.  159 


82 

92 

62 

72 

52 

32 

42 

22 

12 

3 

3 

3 

3 

3 

3 

3 

3 

3 

82 

92 

62 

72 

52 

32 

42 

22 

12 

4 

4 

4 

4 

4 

4 

4 

4 

4 

In  like  manner,  5,  6,  7,  8,  and  9  may  be  used  as  sub- 
trahends. The  exercises  may  be  continued  by  chang- 
ing the  unit  figure  of  the  minuend,  and  taking  each  of 
the  higher  digital  numbers  as  a  subtrahend  number. 

The  following  table  represents  all  of  these  difficulties, 
one  of  each : 


20    20   20 

20 

20 

20 

20 

20 

20 

987 

6 

5 

4 

3 

2 

1 

41   41 

41 

41 

41 

41 

41 

41 

9    8 

7 

6 

5 

4 

3 

2 

52 

52 

52 

52 

52 

52 

52 

9 

8 

7 

6 

5 

4 

3 

73 

73 

73 

73 

73 

73 

9 

8 

7 

6 

5 

4 

84 

84 

84 

84 

84 

_9 

8- 

7 

6 

5 

35 

35 

35 

35 

_9 

8 

7 

6 

56 

56 

56 

9 

8 

7 

67 

67 

9 

8 

Each  of  these  combinations  can  be  readily  expanded 
into  a  series  of  blackboard  drills. 


160  ORAL   LESSONS  IN  NUMBER. 


III.  MULTIPLICATION. 

Blackboard  exercises  in  multiplication  are  easily  made 
by  writing  a  line  of  the  digital  numbers,  and  then  re- 
quiring the  pupils  to  multiply  each  number,  from  right 
to  left,  by  a  given  digital  number.  There  is,  however, 
an  advantage  in  writing  the  two  numbers  to  be  multi- 
plied under  each  other,  that  they  may  be  associated  in 
vision  as  well  as  mentally.  The  several  series  of  com- 
binations are  presented  in  the  following  table: 


1. 

f    5 
{    1 

8 
1 

3 
1 

9 
1 

4 
1 

2 
1 

6 
1 

7 
1 

1 

1 

2, 

I    5 

I   2 

8 
2 

3 
2 

9 

2 

4 
2 

2 
2 

6 
2 

7 

2  • 

1 

2 

3. 

f    5 
1    3 

8 
3 

3 
3 

9 
3 

4 
3 

2 
3 

6 
3 

7 
3 

1 
3 

4. 

(    5 

I   4 

8 
4 

3 

4 

9 
4 

4 
4 

2 
4 

6 
4 

7 
4 

1 

4 

5. 

f    5 
1    5 

8 
5 

3 
5 

9 
5 

4 
5 

2 
5 

6 
5 

7 
5 

1 
5 

6. 

I    5 
1    ^ 

8 
6 

3 
6 

9 
6 

4 
6 

2 
6 

6 
6 

7 
6 

1 

6 

7. 

(    5 

1   7 

8 

7 

3 

7 

9 

7 

4 

-    7 

2 

7 

6 

7 

7 
7 

1 

7 

8. 

I   5 

1   8 

8 

8 

3 

8 

9 

'8 

4 

8 

2 

8 

6 

8 

7 
8 

1 

8 

9. 

i   5 
I   9 

8 
9 

3 
9 

9 
9 

4 
9 

2 
9 

6 
9 

7 
9 

1 
9 

NOTE. — The  above  tables  may  be  used  for  drills  in  division 
requiring  the  pupils  to  divide  each  product  by  each  of  its  two 
factors. 


THIRD- YEAR  COURSE.  161 


IV.   DIVISION. 

The  separate  blackboard  drills  in  division  should 
include  the  division  of  numbers  by  digital  numbers 
which  are  not  their  factors,  as  is  shown  in  the  follow- 
ing table  : 


2) 

3 

9 

5 

11 

7 

9 

13 

15 

17 

19 

3) 

7 

11 

8 

13 

17 

20 

23 

25 

26 

29 

4) 

9 

10 

11 

13 

17 

22 

29 

35 

38 

39 

5) 

11 

12 

24 

36 

38 

41 

43 

46 

48 

49 

6) 

13 

19 

25 

31 

45 

38 

49 

28 

17 

32 

6) 

15 

50 

52 

35 

38 

44 

45 

56 

58 

59 

7) 

15 

19 

23 

27 

30 

37 

40 

45 

50 

57 

7) 

33 

39 

44 

46 

48 

51 

54 

58 

65 

69 

8) 

18 

20 

26 

30 

33 

37 

41 

45 

50 

54 

8) 

47 

43 

47 

58 

61 

63 

66 

70 

75 

79 

9) 

19 

21 

26 

30 

32 

37 

40 

42 

47 

49 

9) 

55 

57 

59 

66 

69 

71 

76 

78 

85 

89 

10) 

32 

43 

56 

64 

78 

82 

96 

77 

88 

99 

The  numbers  in  the  above  table  should  be  divided 
from  left  to  right,  and  the  pupils  should  give  only 
results;  thus,  taking  third  series  for  example,  the  re- 
sults are  :  2  and  1  over ;  2  and  2  over ;  2  and  3  over  ; 
3  and  1  over;  4  and  1  over,  etc. 

Each   series   of    numbers    may   be    increased  at   the 

pleasure  of  the  teacher, 
o.  L.— 14. 


MISCELLANEOUS    LESSONS    AND    SUGGESTIONS. 


UNITED   STATES   MONEY. 

The  more  common  money  units,  as  the  cent  and  the 
dollar,  may  be  taught  as  early  as  the  second  year,  and 
pupils  may  also  be  made  familiar  with  all  the  smaller 
coins  in  common  use,  including  the  cent,  two-cent  piece, 
five-cent  piece  (nickel),  ten-cent  piece  (dime),  twenty-five  cent 
piece  (quarter-dollar),  fifty-cent  piece  (half-dollar),  and 
dollar. 

Such  instruction  should  be  given  incidentally,  and 
the  lessons  should  be  repeated  until  the  pupils  are 
familiar  with  the  different  coins  and  their  comparative 
value. 

The  first  lesson  may  be  devoted  to  the  cent  and  two- 
cent  piece;  the  second,  to  the  half-dime  or  "nickel;" 
the  third,  to  the  dime;  the  fourth,  to  the  quarter-dollar; 
the  fifth,  to  the  half-dollar;  and  the  sixth,  to  the  dollar, 
as  indicated  in  the  following  outlines.  The  actual 
coins  should  be  used. 


1.  How  many  cents  equal  in  value  a  two-cent  piece? 
How  many  cents  in  2  two-cent  pieces?  4  two-cent 
pieces  ?  5  two-cent  pieces  ? 

How  many  cents  in  2  two-cent  pieces  and  1  cent? 
3  two-cent  pieces  and  1  cent?  5  two-cent  pieces  and  1 
cent?  etc. 

(163) 


164  ORAL  LESSONS  IN  NUMBER. 

2.  How  many  cents   in   this   half-dime?     (Present  a 
silver  half-dime  or  a  "nickel.")     How  many  cents  in  3 
half-dimes?     In  5  half-dimes? 

How  many  cents  in  2  half-dimes  and  3  cents?  5 
half-dimes  and  2  cents? 

NOTE. — Multiply  these  questions ;  and,  if  practicable,  let  the 
pupils  "count"  the  pieces  of  money.  Make  up  little  problems 
involving  the  making  of  change,  etc. 

3.  How  many  cents  in  this  dime?     In  2  dimes?     4 
dimes?     6  dimes?     8  dimes?     10  dimes? 

How  many  cents  in  2  dimes  and  1  half-dime?  3 
dimes,  2  half-dimes,  and  4  cents?  etc. 

4.  How   many   cents    in    this    quarter-dollar?      How 
many  cents  in  2  quarter-dollars?     In  3  quarter-dollars? 
In  4  quarter-dollars? 

How  many  cents  in  2  quarter-dollars,  1  dime,  and  2 
half-dimes  ? 

How  many  cents  in  1  quarter-dollar,  2  dimes,  and  4 
half-dimes  ? 

5.  How  many  cents  in  this  half-dollar?     How  many 
cents  in  2  half-dollars? 

How  many  cents  in  1  half-dollar,  1  quarter-dollar, 
and  1  dime? 

How  many  cents  in  1  half-dollar,  2  quarter-dollars, 
and  5  dimes? 

6.  How    many    cents    in    this    dollar?     How    many 
cents  in  2  dollars?     3  dollars?    5  dollars? 

How  many  half-dollars  in  a  dollar?  How  many 
quarter-dollars  in  a  dollar?  How  many  dimes? 

How  much  money  in  1  dollar,  2  half-dollars,  2  quar- 
ter-dollars, and  1  dime? 


Pupils  may  also  be  early  taught  to  write  sums  of 
money,  and  to  add,  subtract,  multiply,  and  divide  sums 
of  money.  Such  instruction  is  best  given  orally,  and 


MISCELLANEOUS  LESSONS.  165 

it  should  be  introduced  at  such  times  in  the  first  three 
years  of  instruction  in  number  as  the  teacher  may 
deem  best.  There  should  be  no  elaborate  explanation 
of  the  decimal  notation  as  applied  to  the  writing  of 
dollars  and  cents,  but  the  pupils  should  be 
shown  how  to  write  dollars  and  cents,  the  $  1.25 

figures  denoting  dollars  being  preceded  by  $  5.18 

$,  called  the  dollar  sign,  and  separated  from  $  8.40 

those  denoting  cents  by  a  period  or  point.  $10.65 

Write    on    the    board,    as    at    the    right,  S25.50 

several    sums    of    money,    and    teach    the  $30.25 

pupils  to  read  the  figures  denoting  dollars  $47.80 

and    those    denoting   cents    separately   and  $52.05 

together.     Dictate   similar   sums  of  money  $71:08 

for  the  pupils  to  write,   and  continue  the 
drill  until  they  can  both  read  and  write  small  sums  of 
money  with  accuracy  and  ease. 


When  the  pupils  reach  and  master  the  "  Written 
Exercises "  on  page  38  of  the  ELEMENTARY  ARITHME- 
TIC, repeat  the  exercises  there  given,  changing  the 
numbers  to  sums  of  money.  They  may  be  dictated,  or 
written  on  the  blackboard,  or  both*.  The  first  fifteen 
exercises  may  be  changed  as  follows  : 


2. 

3, 

4. 

5. 

6. 

7. 

$1.25 

$1.08 

$1.35 

$0.98 

$2.23 

$4.63 

$1.48 

$2.48 

$2.09 

$1.46 

$0.99 

$3.39 

$0.78 

$2.29 

$1.05 

$1.76 

$1.78 

$0.88 

$1.07 

$1.39 

$1.06 

$223 

$1.84 

$1.99 

$0.77 

$0.88 

$0.65 

$1.57 

$1.09 

$1.19 

$1.38 

$1.43 

$0.67 

$1.73 

$1.89 

$1.08 

81.CK 

$1.47 

$2.08 

$0.78 

$0.89 

$1.05 

8, 

9. 

10. 

11.         12,         13. 

14. 

15. 

$1.45 

$2.47 

$3.65     $4.73     $5. 

86     $6.93 

$2.87 

$3.64 

$0.36 

$1.43 

$2.46     $ 

;3.44    $2 

.77     $4.78 

$1.79 

$2.46 

166  ORAL  LESSONS  IN  NUMBER. 

The  pupils  will  readily  change  the  remaining  exer- 
cises on  the  page  in  a  similar  manner;  also,  exercises 
27  to  32,  inclusive,  on  page  31. 

Exercises  in  multiplying  and  dividing  sums  of 
money  may  be  readily  provided  by  changing  the  multi- 
plicands and  dividends  on  pages  54,  56,  and  58  to  dol- 
lars and  cents,  by  taking  the  two  right-hand  figures  in 
each  number  for  cents,  and  the  left-hand  figure  for  dol- 
lars, separating  by  a  period,  and  prefixing  the  dollar 
sign,  when  necessary  so  to  do.  The  pupils  will  make 
these  changes  readily,  and  thus  relieve  the  teacher  of 
the  task  of  writing  the  exercises  on  the  board. 


COMMON  MEASURES. 

The  more  common  measures  may  be  taught  'objectively 
in  connection  with  third-year  lessons  in  number,  but  it 
seems  best  to  make  such  instruction  incidental,  intro- 
ducing it  gradually  to  afford  variety,  and  at  such  times 
as  the  pupils  are  best  prepared  to  receive  it  and  profit 
by  it.  If  deferred  until  the  close  of  the  third  year,  the 

following  lessons  can  be  given  consecutively. 

• 

MEASURES   OF    LIQUIDS. 

The  teacher  should  be  supplied  with  a  gill  measure, 
a  pint  measure,  a  quart  measure,  and  a  gallon  measure. 
A  gill  cup,  a  pint  cup,  a  quart  cup,  and  a  gallon  milk- 
can,  with  a  bucket  of  water,  will  answer  every  purpose. 
These  can  be  easily  procured. 

1,  Teach  the  names  of  each  of  the  measures,  and  take 
the  gill  cup  and  show  the  pupils  how  to  find  how 
many  gills  in  a  pint. 

Count,  children,  and  see  how  many  times  I  fill  this 
little  gill  cup  with  water,  and  pour  it  into  the  pint  cup 
to  fill  it.  How  manv  times?  "  Four  times." 


MISCELLANEOUS  LESSONS.  167 

Now  let  us  see  how  many  times  a  pint  of  water  will 
fill  a  gill  cup.  Count  as  I  fill  the  gill  cup  from  the 
pint  cup.  How  many  times?  "Four  times." 

How  many  gills  make  a  pint? 

2.  Susan  may  now  take  the  pint  cup  and  find  how 
many  times  she  must   fill   it   with  water  and   pour  it 
into    the    quart   cup   to    fill    it.       How    many    times? 
"  Twice." 

Jane  may  find  how  many  times  a  quart  of  water  will 
fill  the  pint  cup.     How  many?     "Twice." 
How  many  pints  make  a  quart? 

3.  Charles  may  now  take  the  quart  cup  and  find  how 
many  times  he  can  pour  it  full  of  water  into  the  gallon 
measure.      How    many?     "Four    times."      The   gallon 
measure  is  full. 

Now  Harry  may  take  the  gallon  of  water  and  find 
how  many  times  it  will  fill  a  quart  cup.  How  many 
times?  "Four  times." 

How  many  gills  make  a  pint?  How  many  pints 
make  a  quart?  How  many  quarts  make  a  gallon? 

TABLE. 

4  gills  are  I  pint. 
2  pints  are  I  quart. 
4  quarts  are  1  gallon* 

4.  How   many    pint   bottles   will   hold   3  quarts  of 
vinegar? 

How  many  times  must  you  fill  a  pint  cup  to  measure 
2  quarts  of  milk? 

A  milkman  has  a  three-gallon  can  full  of  milk : 
how  many  quarts  of  milk  can  he  sell?  When  he  sells 
12  quarts,  how  many  .quarts  will  be  left  in  the  can? 

A  housekeeper  filled  6  two-quart  cans  and  8  quart 
cans  with  peaches:  how  many  quarts  of  peaches  did 
she  have?  How  many  gallons? 


168  ORAL  LESSONS  IN  NUMBER 


DRY    MEASURES. 

The  teacher  should  be  supplied  with  a  pint  measure, 
a  quart  measure,  a  peck  measure,  and  at  least  a  peck 
of  some  kind  of  grain  or  small  fruit.  These  can  be 
easily  procured  for  temporary  use.  It  is  very  im- 
portant that  the  pupils  see  and  handle  these  different 
measures ;  and,  when  practicable,  a  bushel  measure. 

The  first  step  is  to  teach  the  several  measures,  and 
their  names,  and  the  next  step  is  to  teach  the  number 
of  pints  in  a  quart,  quarts  in  a  peck,  and  pecks  in  a 
bushel. 


1.  John  may  take  the  pint  measure  and  measure  2 
pints  of  grain ;  4  pints. 

Charles  may  take  the  quart  measure  and  measure  3 
quarts  of  grain ;  6  quarts. 

Mary  may  take  the  pint  measure  and  find  how  many 
pints  of  grain  will  fill  a  quart  measure.  How  many? 
"Two  pints." 

Nellie  may  take  a  quart  of  grain  and  find  how  many 
times  it  will  fill  a  pint  measure.  How  many  ?  "  Two 
times." 

Then  how  many  pints  make  a  quart? 

2.  Clarence   may   take   the   quart  measure   and    find 
how   many   quarts   of  grain   will   fill   a  peck   measure. 
How  many?     "Eight  quarts." 

Now  John  may  find  how  many  times  a  peck  of  grain 
will  fill  a  quart  measure.  How  many  ?  u  Eight  times." 

Then  how  many  quarts  make  a  peck? 

If  we  had  a  bushel  measure  here,  it  would  take  4 
pecks  of  grain  to  fill  it.  How  many  pecks  make  a 
bushel  ? 

3.  How  many  pint  baskets  can  I  fill  with  one  quart 
of  cherries? 


MISCELLANEOUS  LESSONS.  169 

How  many  quart  bags  can  I  fill  with  a  peck  of  nuts  ? 

How  many  peck  measures  can  I  fill  with  a  bushel  of 
wheat  ? 

A  boy  picked  2  pecks  of  chestnuts,  and  sold  them  at 
10  cents  a  quart :  how  many  quarts  of  nuts  did  he  sell  ? 

How  many  pint  bottles  will  5  quarts  of  seed  fill? 

TABLE. 

2  pints  are  1  quart. 
8  quarts  are  1  peck. 
4  pecks  are  1  bushel. 


MEASURES   OF    LENGTH   OR   DISTANCE. 

The  teacher  should  be  supplied  with  a  foot-rule,  a 
yard-stick,  and  a  piece  of  tape  at  least  a  rod  long,  and 
accurately  divided  into  feet.  Each  pupil  should  be 
supplied  with  a  foot-rule,  or  with  a  narrow  strip  of 
strong  paper,  one  foot  in  length,  and  plainly  divided 
into  inches. 

1.  The  first  step  is  to  develop  the  idea  of  length  or 
distance  by  comparing  objects  of  different  lengths,  and 
training  pupils  in  estimating  the  length  of  objects  "by 
the  eye." 

Hold  up  two  objects  nearly  of  the  same  length,  and 
have  the  pupils  judge  which  is  the  longer.  Test  by 
putting  the  objects  close  together. 

Draw  two  lines  on  the  blackboard  of  different  lengths 
and  in  different  positions,  and  have  pupils  judge  which 
is  the  longer.  Test  by  measuring  lines  with  the  rule. 

Draw  horizontal  lines  from  one  to  twelve  inches  long, 
and  have  pupils  estimate  their  length  in  inches.  Test 
by  applying  the  rule. 

Draw  vertical  and  oblique  lines  on  the  board,  and 
have  their  lengths  estimated  by  the  pupils  and  then 

measured  with  the  rule, 
o.  L.— 15. 


170  ORAL   LESSONS  IN  NUMBER. 

Have  pupils  estimate  the  length  and  width  of  panes 
of  glass,  of  slates  of  different  sizes,  books,  etc.,  and  then 
test  accuracy  by  applying  the  rule. 

2.  In  like  manner,  the  length  of  lines,  as  the  length 
and  width  of  the  table,  blackboard,  etc.,  may  be  esti- 
mated  in   feet   and   measured,   and   the  length   of   the 
sides  of  the  room,  long  strings,  etc.,  may  be  estimated 
in  yards  and  measured. 

The  rod  measure  may  also  be  introduced,  and  the 
length  and  width  of  the  school-yard,  the  width  of  the 
street,  etc.,  may  be  measured  by  a  line  one  rod  in 
length. 

3.  This  is  a  foot-rule :    how  long  is  it  ?     "  One  foot." 
How  many  inches  long  is  it?     "Twelve  inches." 

Here  is  a  line  just  24  inches  long:  how  many  feet 
long  is  it? 

Here  is  a  string  1  yard  long :  if  I  cut  it  into  pieces, 
each  1  foot  long,  how  many  pieces  will  it  make? 

How  many  feet  make  a  yard  ? 

Here  is  a  board  9  feet  long:  what  is  its  length  in 
yards  ? 

How  many  feet  in  a  ribbon  4  yards  long? 

TABLE. 

12  inches  are  I  foot. 
3  feet  are  1  yard. 
5^  -yards  are  1  rod. 


MEASURES   OF    TIME. 


1.  The  portion  of  time  called  a  day  begins  at  12 
o'clock  at  night,  or  at  midnight,  and  ends  at  12  o'clock 
the  next  night. 

When  did  to-day  begin  ?  "  At  12  o'clock  last  night." 
When  will  to-day  end  ?  "  At  12  o'clock  to-night" 


MISCELLANEOUS  LESSONS.  171 

What  is  12  o'clock  at  night  called?  "It  is  called 
midnight."  How  long,  then,  is  a  day?  "From  mid- 
night to  the  next  midnight." 

What  is  the  middle  of  the  day  called  ?  "  It  is  called 
midday,  or  noon."  The  noon  divides  the  day  into  how 
many  equal  parts?  "Two  equal  parts." 

2.  A   day   is   divided    into    twenty-four   equal   parts 
called  hours.     How  many  hours  from  midnight  to  mid- 
night?    From  midnight  to  noon?     From  noon  to  mid- 
night? 

When  does  the  clock  strike?  "At  the  end  of  each 
hour."  How  many  times  does  the  clock  strike  in  a 
a  day?  "Twenty-four  times."  How  many  times  does 
it  strike  from  midnight  to  noon  ?  From  noon  to  mid- 
night ? 

How  many  hours  in  a  day? 

3.  An   hour  is  divided  into  sixty  equal  parts  called 
minutes.     The  pupils  may  all  keep  still  for  one  minute. 
How  many   times  this  silence  would   make  an  hour? 
"Sixty  times."     How  many  minutes  in  the  school  re- 
cess ?     How  many  minutes  in  this  class  exercise  ? 

How  many  minutes  in  an  hour? 

Put  your  finger  on  your  pulse  and  count  silently 
sixty  "  beats."  It  has  taken  a  little  less  than  a  minute. 
Count  to  sixty  as  I  beat  seconds  with  my  hand.  (This 
may  be  done  by  the  guidance  of  the  second-hand  of  a 
watch.)  How  long  has  it  taken?  "One  minute." 

How  many  seconds  in  a  minute? 

TABLE. 

60  seconds  are  1  minute. 
60  minutes  are,  1  hour. 
24  hours  are  1  day. 

The  teacher  may  also  teach  the  number  of  days  in 
a  week;  the  names  of  the  days  of  the  week  in  their 
order;  the  number  of  days  in  a  year;  the  number  of 


172  ORAL  LESSONS  IN  NUMBER. 

months  in  a  year;  the  names  of  the  months  in  their 
order ;  the  division  of  the  year  into  four  seasons ;  the 
names  of  the  months  in  each  season,  etc. 


MEASURES  OF  WEIGHT. 

In  teaching  weight,  the  teacher  should  be  supplied 
with  a  common  balance,  the  usual  weights,  and  several 
parcels,  each  weighing  one  pound. 

1.  The  first  step  is  to  give  the  pupils  an  idea  of  the 
weight  called  a  pound.     Let  the  pupils  take  a  pound 
parcel  in  one  hand,  and  a  pound  weight  in  the  other, 
and  "  heft "  them.     Tell  them  that  a  pint  of  cold  water 
weighs  a  pound  (nearly).*    Let  the  pupils  "heft"  two- 
pound  parcels,   three-pound  parcels,  etc. 

2.  Teach  the  division  of  the  pound  into  sixteen  equal 
parts   called   ounces.     Show    an    ounce    weight,    a   four- 
ounce  weight,  and  an  eight-ounce  weight.     Let  the  pu- 
pils weigh  different  articles. 

How  many  ounce  weights  weigh  as  much  as  a  pound 
weight?  How  many  four-ounce  weights  are  a  pound? 
How  many  eight-ounce  weights? 

What  part  of  a  pound  is  an  ounce?  What  part  of 
a  pound  are  four  ounces?  Eight  ounces? 

3.  Teach  the  pupils  that  one  hundred  pounds  is  a 
hundred-weight,   and  that  t \venty  hundred-weights  are 
a  ton. 

How  many  pounds  in  a  hundred-weight  of  hay? 
Three  hundred-weights? 

How  many  pounds  in  a  ton?  In  two  tons?  Four 
tons  ?  One  half  of  a  ton  ?  One  fourth  of  a  ton  ? 


*A  gallon  of   ice-cold   water  weighs   a   little  more   than 
pounds,  and  hence  a  pint  weighs  1^  pounds. 


MISCELLANEOUS  LESSONS.  173 


TABLE. 

16  ounces  are  1  pound. 
100  pounds  are  I  hundred-weight. 
20  hundred-weights  are  1  ton. 


METRIC  MEASURES. 

Since  the  metric  measures  are  not  in  common  use, 
it  is  not  necessary  to  teach  them  as  a  part  of  the  ele- 
mentary course  in  number.  Pupils  should  first  be 
made  familiar  with  the  common  measures. 

When  the  metric  measures  are  introduced,  care  should 
be  taken  to  make  the  pupils  practically  familiar  with 
the  metric  units,  and  this  can  best  be  accomplished  by 
the  actual  use  of  the  metric  measures  in  measuring.  To 
this  end,  the  school  should  be  supplied  with  a  meter 
measure,  a  liter  measure,  and  gram  and  kilogram 
weights.  When  pupils  learn  that  a  new  nickel  five 
cent  piece  wreighs  5  grams,  and  two  of  them  10  grams, 
they  will  soon  have  a  very  clear  idea  of  the  gram. 

The  pupils  should  have  frequent  practice  in  the  use 
of  these  measures.  With  the  meter  they  should 
measure  the  length  and  width  of  the  school-room  floor, 
the  teacher's  desk,  the  blackboard,  etc. ;  also  the  dis- 
tance between  objects  in  the  school-room,  in  the  school- 
yard, etc.  With  the  liter  they  should  measure  water, 
grain,  etc.,  and  with  the  gram  and  kilogram  they  should 
weigh  different  articles. 

These  exercises  in  metric  measurements  should  be 
introduced  one  or  two  years  before  the  systematic 
teaching  of  the  metric  system  as  presented  in  the  NEW 
COMPLETE  ARITHMETIC.  It  is  only  by  long  practice 
that  pupils  can  be  made  as  familiar  with  the  metric 
measures  as  they  are  with  the  common  measures,  and 


174  ORAL  LESSONS  IN  NUMBER. 

all  such  instruction  must  be  given  orally.  It  is  a  mistake 
to  cumber  the  pages  of  an  elementary  text-book  for 
pupils  with  such  introductory  and  preparatory  drills. 

In  these  preparatory  lessons  no  attention  should  be 
given  to  the  metric  equivalents,  and  the  only  com- 
parisons made  between  the  metric  measures  and  the 
common  measures  should  be  by  the  eye.  By  placing  a 
meter  and  a  yard-stick  together,  the  pupils  will  see  that 
the  meter  is  a  little  longer  than  the  yard  ;  by  pouring 
a  liter  of  water  or  of  grain  into  a  dry  quart  measure, 
and  also  into  a  liquid  quart  measure,  the  pupils  will 
see  the  liter  is  a  little  less  than  the  dry  quart,  and  a 
little  more  than  the  liquid  quart ;  and  by  placing  a 
kilogram  weight  in  one  pan  of  a  balance,  and  a  two- 
pound  weight  in  the  other  pan,  they  will  see  that  a 
kilogram  is  a  little  more  than  two  pounds.  They  will 
thus  learn  and  can  easily  remember  that  the  meter  is  a 
little  more  than  a  yard  ;  the  liter, about  a  quart;  and  a 
kilogram,  a  little  more  than  two  pounds. 

It  Avill  be  time  enough  to  teach  the  exact  numerical 
equivalents,  when  pupils  are  so  familiar  with  the  metric 
measures  that  they  can  think  of  them  without  any 
reference  to  the  common  measures.  When  a  pupil  is 
told,  for  example,  that  a  room  is  8  meters  long  and  5 
meters  wide,  he  should  be  able  to  comprehend  its  di- 
mensions without  reducing  them  to  yards  or  feet;  and 
this  result  can  only  be  attained  by  the  continued  use 
of  the  meter  in  measuring  distances. 

The  reduction  of  the  numbers  composed  of  metric 
units  to  equivalent  numbers  composed  of  common 
units,  and  vice  versa,  is  the  final  step  in  teaching  the 
metric  system.  The  early  introduction  of  the  metric 
equivalents  and  the  reductions  of  metric  numbers  to 
like  common  denominate  numbers,  are  mistakes  which 
have  resulted  in  much  confusion. 


MISCELLANEOUS  LESSONS.  175 


SUBTRACTION  :  WRITTEN   PROCESS. 

There  are  two  methods  or  processes  of  subtracting 
one  number  from  another  when  a  term  of  the  subtra- 
hend is  greater  than  the  corresponding  term  of  the 
minuend.  These  processes  are  as  follows : 

1.  The  adding  of  10  to  the  term  of  the  minuend,  and 
then   subtracting  1   from  the  next  higher  term  of  the 
minuend,  or  considering  it  1  less. 

Thus,  in  subtracting  487  from  659,  ten  (10)  659 
is  added  to  the  5  (making  15),  and  1  is  sub-  487 
tracted  from  the  6  (leaving  5) ;  that  is,  the  6  172 
is  considered  5. 

2.  The  adding  of  10  to  the  term  of  the  minuend,  and 
1    to  the   next   higher  term  of  the   subtrahend. 

Thus,  in  subtracting  487  from  659,  10  is  659 
added  to  the  5  (making  15),  and  1  is  added  to  487 
the  4  (making  5).  172 

There  are  two  methods  of  explaining  each  of 
these  processes,  as  follows : 

The  first  process  may  be  explained  by  showing  that 
the  adding  of  10  to  a  term  of  the  minuend,  and  sub- 
tracting 1  from  the  next  higher  term,  increases  and  de- 
creases the  minuend  equally,  and  hence  its  numerical 
value  is  not  changed.  The  subtracting  of  1  from  the 
next  higher  term  of  the  minuend  offsets  the  10  added 
to  the  lower  term. 

The  first  process  may  also  be  explained  by  showing 
that  the  10  added  to  the  term  of  the  minuend  is  ob- 
tained by  taking  1  from  the  next  higher  term  and  re- 
ducing it  to  10  of  the  next  lower  order.  Instead  of 
changing  the  higher  minuend  figure,  the  term  is  con- 
sidered 1  less;  that  is,  1  is  mentally  subtracted  from  it. 

NOTE. — When  the  next  higher  term  of  the  minuend  is  0,  1 
is  taken  from  the  next  higher  term  whose  value  is  one  or  more, 
and  the  necessary  reductions  to  lower  orders  are  made  until  10 


176  ORAL  LESSONS  IN  NUMBER. 

is  obtained  to  increase  the  minuend  term.  Then  each  0  of  the 
minuend  is  considered  9,  and  the  first  significant  figure  of  it 
reached  is  considered  1  less.  This  second  explanation  seems  to 
be  a  favorite  one  with  teachers,  though  few,  if  any,  account- 
ants make  such  reductions  when  subtracting  numbers. 

The  second  process  may  be  explained  by  showing 
that  the  adding  of  10  to  a  term  of  the  minuend,  and  1 
to  the  next  higher  term  of  the  subtrahend,  increases 
both  minuend  and  subtrahend  equally,  and  hence  their 
difference  is  not  changed. 

This  may  be  illustrated  by  taking  any  two  numbers, 
as  7  and  4,  and  adding  say  8  to  each.  The  difference 
between  7  and  4  is  3,  and  the  difference  between  15 
and  12  is  3. 

The  second  process  may  also  be  explained  by  sup- 
posing that  the  10  added  to  the  minuend  is  obtained 
by  taking  1  from  the  next  higher  term  of  the  minuend, 
and  then  showing  that,  instead  of  changing  the  higher 
minuend  figure,  or  considering  the  term  1  less  (that 
is,  subtracting  mentally  1  from  it),  the  1  may  be  added 
to  the  higher  term  of  the  subtrahend  before  subtract- 
ing, and  thus  the  1  and  the  subtrahend  term  be  both 
taken  from  the  higher  minuend  term  at  the  same  time. 

This  may  be  illustrated  by  subtracting  487  from  659. 
The  10  added  to  the  5  (tens)  may  be  considered  as  ob- 
tained from  the  6  hundreds;  but,  instead  of  subtract- 
ing the  1  from  the  6  (considering  6  diminished  by  1), 
and  then  taking  4  from  5,  both  the  1  and  the  4  may 
be  taken  from  the  6  by  one  subtraction  by  taking  the 
sum  of  the  1  and  the  4,  which  is  5,  from  the  6. 

It  is  thus  seen  that  in  the  first  process  two  subtrac- 
tions are  made,  while  in  the  second  process  the  1  is 
added  to  the  subtrahend  term,  and  but  one  subtraction 
is  made. 

The  term  "  borrow "  should  not  be  used  in  the  de- 
scription or  explanation  of  either  process.  The  10 


MISCELLANEOUS  LESSONS.  177 

added  to  the  minuend  figure  is  not  obtained  by  "  bor- 
rowing." 

There  is  a  difference  of  opinion  among  arithmeticians 
respecting  the  comparative  merits  of  these  two  pro- 
cesses. The  writer  uses  and  prefers  the  second  process, 
possibly  the  result  of  early  training.  It  seems  to  him 
more  natural  and  more  easily  explained  than  the  first 
process.  The  difference,  however,  is  so  slight  that 
pupils  who  have  learned  either  process  should  not  be 
required  to  learn  the  other. 

NOTE. — There  is  an  obvious  advantage  in  using  the  word 
"term,"  in  descriptions  of  elementary  processes,  to  denote  the 
value  of  each  of  the  successive  figures  which  express  a  number. 
It  avoids  the  alternative  of  an  incorrect  use  of  the  word  "  figure," 
or  an  awkward  use  of  the  phrase  "  the  number  denoted  by  each 
figure,"  or  its  modified  equivalent.  A  figure  is  not  a  number. 
Figures  can  not  be  added,  subtracted,  multiplied,  or  divided. 
The  numbers  denoted  by  figures  are  added,  subtracted,  etc. 

The  advantage  of  using  the  word  term,  instead  of  the  expression, 
the  number  denoted  by  a  figure,  is  shown  by  substituting  the  latter 
phrase,  with  needed  modification,  for  the  word  term  as  used  in 
the  above  descriptions  of  the  process  of  subtraction.  For  exam- 
ple, the  expression  "  when  a  term  of  the  subtrahend  is  greater 
than  the  corresponding  term  of  the  minuend,"  becomes  "  when 
the  number  denoted  by  a  figure  of  the  subtrahend  is  greater 
than  the  number  denoted  by  the  corresponding  figure  of  the 
minuend."  There  is  no  justification  of  the  use  of  the  word 
figure  for  term  in  such  expressions. 

The  use  of  the  word  term  to  designate  a  part  of  a  number 
is  analagous  to  its  subsequent  use  in  arithmetic  and  algebra,  as 
in  the  expressions,  the  terms  of  a  compound  number,  the  terms  of  a 
fraction,  the  terms  of  a  ratio  or  proportion,  the  terms  of  a  series,  the 
terms  of  a  polynomial,  etc. 


178  ORAL   LESSONS  IN  NUMBER. 

DIVISION. 

SHORT    AND    LONG    DIVISION. 

It  is  the  practice  in  some  schools  to  teach  long  divi- 
sion before  short  division.  It  is  claimed  that  the  writ- 
ing of  the  partial  products  and  dividends,  as  is  done  in 
long  division,  makes  the  process  of  division  easier  than 
it  is  when  these  successive  partial  products  and  divi- 
dends are  kept  in  mind,  as  in  short  division. 

This  may  be  true,  and  yet  the  training  which  the 
pupils  have  received  has  so  prepared  them  for  this 
mental  process  that  it  presents  little,  if  any,  difficulty. 
If  this  be  not  true,  the  pupils  need  this  very  training. 
Besides,  short  division  is  nearly,  if  not  quite,  as  easily 
mastered  before  as  after  long  division,  while  its  previous 
mastery  assists  the  pupil  in  learning  long  division — 
certainly  the  more  difficult  process  when  the  divisor 
contains  two  or  more  figures.  But  whether  short  or 
long  division  be  first  taught,  only  digital  numbers 
should  be  used  for  divisors  until  the  process  of  short 
division  is  mastered. 

Place  of  the  Quotient. 

The  difference  between  short  and  long  division  is  best 
shown  by  writing  the  quotient  in  each  process  above  the 
dividend,  as  in  the  examples  below : 

SHORT  DIVISION.  LONG  DIVISION. 

672,   Quotient.  672,   Quotient. 


7)4704,  Dividend.  7)4704,  Dividend. 

42_ 

50 


14 

14 


MISCELLANEO  US  LESSONS.  179 

SHORT  DIVISION.  LONG  DIVISION. 

392,   Quotient.  392,   Quotient. 

12)4704,  Dividend.  12)4704,  Dividend. 

36_ 

110 

108 


24 
24 

It  is  thus  clearly  shown  that  the  only  difference  be- 
tween the  two  processes  is  that  the  partial  products  and 
partial  dividends  are  formed  mentally  and  kept  in 
mind  in  short  division,  while  they  are  written  in  long 
division.  The  pupil  should  be  required  to  use  short 
division  when  the  divisor  is  10  or  less. 

The  preceding  method  of  writing  the  quotient  in 
long  division  is  used  by  some  teachers.  If  adopted  in 
long  division,  there  will  be  an  advantage  in  using  it 
also  in  short  division,  thus  giving  the  quotient  the 
same  position  in  both  processes. 

If  the  quotient  be  written  above  the  dividend  in  the 
division  of  integral  numbers,  it  should  also  be  written 
above  in  the  division  of  decimals. 

There  is,  however,  good  reason  for  the  almost  uni- 
versal practice  of  writing  the  quotient  below  the  divi- 
dend in  short  division,  and  at  the  right  in  long  division. 
In  the  solution  of  problems,  it  often  becomes  necessary 
to  divide  a  sum,  difference,  or  product,  and  this  can 
best  be  done  by  writing  the  quotient  under  or  at  the 
right.  The  rewriting  of  the  dividend,  to  make  room 
for  the  writing  of  the  quotient  above  it,  would  break 
the  continuity  of  the  written  solution,  as  well  as 
lengthen  the  process.  The  reduction  of  denominate 
numbers  to  a  lower  denomination,  and  the  solution  of 
problems  in  partial  payments,  when  the  rate  is  more  or 
less  than  six,  afford  good  illustrations  of  this  difficulty. 


180  ORAL  LESSONS  IN  NUMBER. 


The  Determining  of  Quotient  Figures. 

The  principal  difficulty  in  long  division  is  in  deter- 
mining the  successive  figures  of  the  quotient,  especially 
when  the  divisor  is  a  large  number.  This  difficulty 
may  be  greatly  lessened  by  the  following  method : 

1.  Begin   with   divisors   that   are   expressed    by    one 
figure. 

2.  Next  take  examples  in  which  each  successive  quo- 
tient figure  is  found  by  dividing  the  left-hand  term  of  each 
successive  partial   dividend    by   the    left-hand    term   of   the 
divisor. 

3.  Then   take  examples  in  which  the  two  left-hand 
terms  are  1  and  0  respectively,  thus  making  10  the  trial 
divisor. 

The  following  solutions  illustrate  these   three  steps  : 

(1)  (2)  (3) 

8)5088(636  224)51968(232  1036)452732(437 

48  448  4144 

28  716  3833 

24  672  3108 


48  448  7252 

48  448  7252 


In  the  second  solution,  the  successive  quotient  figures 
are  found  by  dividing  5,  7,  and  4  respectively  by  2,  the 
left-hand  term  of  the  divisor,  and  in  the  third  solution 
the  successive  quotient  figures  are  found  by  dividing 
45,  38,  and  72  respectively  by  10,  the  number  expressed 
by  the  two  left-hand  figures  of  the  divisor. 

By  solving  several  examples  of  each  of  these  three 
kinds,  pupils  will  not  only  become  familiar  with  the 
long  division  process,  but  they  will  be  prepared  to  use 


MISCELLANEOUS  LESSONS.  181 

as  a  trial  divisor  a  number  expressed   by   one,  two,  or 
three  of  the  left-hand  figures  of  the  divisor. 

NOTE. — The  first  fifteen  problems  in  long  division  in  the  ELE- 
MENTARY ARITHMETIC  (page  105)  are  arranged  in  accordance  with 
the  above  method. 


ORAL  SOLUTIONS. 

In  the  lessons  in  number  given  the  first  three  years, 
no  attempt  should  be  made  to  teach  the  .logical  analysis 
of  problems,  and  generally  nothing  is  gained  by  stating 
formally  the  reasons  for  processes  and  results. 

In  solving  the  numerous  little  problems  which  em- 
body the  applications  of  processes,  the  pupils  should 
first  give  the  answer,  and  then  state  the  process  or 
processes  by  which  it  is  reached,  as  follows : 

(1)  2  pears  :  9  pears  less  7  pears  are  2  pears. 

(2)  13  cents :  8  cents  and  5  cents  are  13  cents. 

(3)  15  cents :  5  times  3  cents  are  15  cents.* 

(4)  4  rows :  4  trees  are  contained  in  16  trees  4  times. 
The  oral  solutions  of  problems  in  the  fourth  and  fifth 

years  should  also  be  concise  and  simple.  Young  pupils 
are  not  helped  by  an  attempt  to  give  a  minute  and 
formal  statement  of  every  condition  involved  in  a  prob- 
lem ;  and,  at  no  stage  of  their  advancement,  is  the 
reasoning  faculty  trained  by  the  repetition  of  what  has 


*  It  is  absurd  to  require  young  children  to  give  such  a  solu- 
tion as  this: 

Teacher.— "  What  will  five  apples  cost  at  three  cents  apiece  ?" 

Pupil— "  Five   apples   will   cost   five   times  as   much  as  one 

apple.     Hence,  if  one  apple  costs  three   cents,  five   apples  will 

cost  five  times  three  cents,  which   is  fifteen   cents.     Therefore, 

if  one  apple  costs  three  cents,  five  apples  will  cost  fifteen  cents." 


182  ORAL  LESSONS  IN  NUMBER 

been  aptly  called  "  logical  verbiage."  It  is  now  ad- 
mitted that  the  elaborate  logical  analyses  of  problems 
which  pupils  were  formerly  required  to  give  in  what  is 
called  "  mental  arithmetic,"  was  a  serious  hindrance  to 
the  mastery  of  the  processes  and  principles  of  arithme- 
tic, and  it  is  equally  evident  that  it  was  an  injury  to 
the  thinking  power  of  children.  Much  of  the  glibbest 
logical  analysis,  once  the  pride  of  so  many  teachers, 
was  the  result  of  the  worst  form  of  rote  teaching,  the 
analyses  being  committed  to  memory  by  the  pupils, 
and  repeated  without  any  wholesome  exercise  of  the 
logical  faculty. 

This  wide  abuse  of  the  so-called  "mental  arithmetic" 
has  led  many  teachers  to  underestimate  the  value  of 
analytic  drills  in  teaching  arithmetic ;  and,  as  a  conse- 
quence, they  have  a  small  place,  if  any,  in  their  in- 
struction. The  truth  is,  the  clear  analysis  of  problems 
has  a  very  important  place  in  arithmetical  instruction, 
and  hence  the  so-called  "  oral  problems "  in  the  arith- 
metics used  should  be  taught  with  as  much  thorough- 
ness as  the  written  problems,  especial  care  being  always 
taken  to  adapt  the  form  of  analysis  required  to  the 
capacity  and  advancement  of  the  pupils.  The  oral 
problems  in  the  higher  book  are  not  only  more  difficult 
than  those  in  the  lower  book,  but  the  analyses  are 
properly  more  logical  and  formal.  A  careful  study  of 
the  model  solutions  given  in  the  ELEMENTARY  ARITH- 
METIC, will  assist  teachers  in  avoiding  the  error  referred 
to  above. 


WRITTEN   PROCESSES. 

All  the  written  processes  in  the  elementary  course 
should  be  taught  inductively.  The  oral  exercises 
which  precede  the  written  problems,  are  often  so  com- 
plete an  introduction  to  the  corresponding  written  pro- 


MISCELLANEOUS  LESSONS.  183 

cesses  that  pupils  can  pass  from  one  to  the  other 
without  difficulty.  All  that  is  necessary,  in  most  cases, 
is  to  put  on  the  blackboard  the  written  solutions  of 
two  or  three  of  the  oral  problems  in  connection  with 
the  oral  solutions.  But  this  step  should  not  be  taken 
until  the  inductive  oral  exercises  have  all  been  recited  orally. 
The  pupils  should  first  master  the  oral  processes,  and 
then  be  led  to  pass  from  or  through  these  to  the  written 
processes. 

When  the  inductive  problems  have  been  solved  both 
orally  and  on  the  slate,  the  pupils  are  prepared  to  pass 
to  the  solution  of  the  so-called  "  written  problems." 
All  the  written  problems,  assigned  for  a  lesson,  should 
be  solved  by  the  pupils  on  slate  or  paper,  and  the  solu- 
tions should  be  brought  to  the  recitation  for  the  teacher's 
inspection  and  approval.  The  solutions  should  be  made 
in  an  approved  form,  and  they  should  be  arranged  in  a 
neat  and  systematic  manner.  A  little  instruction  will 
enable  pupils  to  make  an  economic  use  of  space  in 
slate  and  blackboard  work,  and,  at  the  same  time, 
present  each  solution  in  an  intelligible  form.  When 
the  solutions  of  problems  are  properly  arranged  and 
written,  two  to  three  minutes  will  suffice  to  inspect  the 
slate  work  of  a  large  class,  and  often  this  may  be  done 
in  less  than  a  minute.  The  accuracy  and  the  neatness 
of  the  written  solutions  should  receive  attention. 


RULES. 

The  old  method  of  teaching  arithmetical  processes  by 
requiring  pupils,  first,  to  commit  to  memory  a  formal 
rule,  and  then  to  solve  the  problems  "according  to  the 
rule,"  and  with  constant  reference  to  it,  was  long  since 
discarded  by  the  most  successful  teachers.  Experience 
has  shown  that  the  rule  is  not  only  useless  as  a  means 


184  ORAL  LESSONS  IN  NUMBER. 

of  teaching  numerical  processes,  but  that  it  is  an  actual 
hindrance.  It  has  also  shown  that  a  knowledge  of  the 
process  is  essential  to  the  proper  teaching  of  the  rule. 
Hence,  "processes  before  rules"  and  "rules  through  pro- 
cesses," have  been  generally  accepted  as  wise  maxims  for 
the  teaching  of  elementary  arithmetic. 

Since  the  rule  is  to  be  taught  after  the  process,  the 
author's  rule  should  be  placed  after  the  problems  in  all 
elementary  arithmetics.  The  placing  of  the  author's 
rule  before  the  problems,  or  after  the  first  four  or  five 
problems,  is  a  violation  of  the  true  order  of  teaching 
processes  and  rules,  and  it  leads  to  an  improper  use  of 
the  rule  in  the  solution  of  the  remaining  problems. 
Whether  all  the  problems  should  be  solved  before  a 
rule  is  generalized,  is  a  question  for  each  teacher  to  de- 
cide, but  the  author's  statement  of  the  rule  is  best 
taught  after  the  given  problems  have  been  solved  and 
the  process  thoroughly  mastered.  In  teaching  any  pro- 
cess, the  successive  steps  will,  of  course,  be  taught,  and 
the  pupils  will  be  required  again  and  again  to  describe 
these  steps  in  words,  but  all  this  will  be  done  with 
direct  reference  to  the  mastery  of  the  processes  as  such. 
Attention  will  also  be  given  to  the  primary  principles 
involved,  but  this  will  not  be  chiefly  directed  to  their 
concise  statement  in  language.  In  other  words,  the 
primary  facts  involved  in  principles  and  rules  should 
be  taught  incidentally  in  connection  with  the  solution 
of  the  problems.  The  teaching  of  principles  and  rules 
inductively,  with  the  memorizing  of  their  best  possible 
statement,  is  the  final  step. 

When  the  formal  rule  is  taught,  it  should  be  derived 
from  the  process  by  the  pupils  under  the  guidance  of 
the  teacher.  The  trae  order  of  the  successive  steps  is 
as  follows  : 

1.  A  mastery  of  the  process  without  reference  to  the 
author's  rule. 


MISCELLANEOUS  LESSONS.  185 

2.  The  recognition  and  statement  of    the  successive 
steps  of  the  process  in  their  order. 

3.  The  combination  of  these  several  statements  into 
a  general  statement. 

4.  A  comparison  of  the  general  statement  thus  formed 
with  the  author's  rule. 

5.  The  memorizing  of  the  approved  rule. 

The  rule  for  addition,  for  illustration,  may  thus  be 
taught : 

What  is  the  first  step  in  the  addition  of  numbers? 
"  To  write  the  numbers  to  be  added." 

How  are  the  numbers  to  be  written?  "So  that  the 
figures  which  denote  the  same  order  of  units  shall  be 
in  the  same  column." 


NOTE. — Take  an  example,  and  solve  it  on  the  blackboard  by 
doing  just  what  each  answer  describes.  Lead  the  pupils  to  cor- 
rect all  such  wrong  expressions  as  "  figures  of  the  same  order," 
the  writing  of  the  figures  "under  each  other,"  etc.  The  correct 
expressions  have,  of  course,  been  previously  used  in  describing 
solutions  of  problems. 


What  is  the  next  step?    "Draw  a  line  underneath." 

Now  put  your  three  statements  together  in  one  sen- 
tence, and  I  will  write  it  on  the  blackboard  : 

"  Write  the  numbers  to  be  added  so  that  figures  which 
denote  the  same  order  of  units  shall  be  in  the  same  column, 
and  draw  a  line  underneath" 

What  is  the  next  step  ?  "  Add  the  numbers  in  each 
column." 

With  which  column  do  you  begin  ?  "  With  the 
units'  column." 

What  is  done  with  the  sum  of  each  column  ?  "  Write 
the  sum  underneath  the  column  added." 

When  can  this  be  done  ?  "  When  the  sum  is  less 
than  ten." 

O.  L.-16. 


18ft  ORAL  LESSONS  IN  NUMBER. 

Now  put  these  four  statements  in  one  sentence,  and 
I  will  write  it  on  the  blackboard : 

"  Beginning  with  units'  column,  add  the  numbers  in  each 
column,  and  write  the  sum,  when  less  than  ten.  underneath" 

What  is  done  when  the  sum  of  the  numbers  in  any 
column  is  ten  or  more  ? 

"  When  the  sum  of  any  column  is  ten  or  more,  write  the 
right-hand  figure  under  the  column  added,  and  add  the 
number  denoted  by  the  left-hand  figure  or  figures  With  the 
next  column." 

I  have  written  this  statement  on  the  blackboard. 

How  is  the  sum  of  the  numbers  in  the  left-hand 
column  written? 

"  Write  the  entire  sum  of  the  left-hand  column,  placing 
the  right-hand  figure  under  the  column  added" 

These  four  sentences  make  the  rule  for  addition. 

When  the  rule  has  thus  been  developed  and  cor- 
rected, have  the  several  steps  repeated  in  connection 
with  the  solution  of  a  problem  on  the  board,  one  pupil 
giving  the  first  sentence,  another  the  second,  and  so  on. 
Finally,  perfect  the  rule  by  comparing  it  with  the 
author's  rule,  and  then  require  the  pupils  to  repeat  the 
entire  rule  with  accuracy  and  without  hesitation. 


DEFINITIONS. 

The  definitions  should,  in  like  manner,  be  taught 
inductively,  and  they  should  first  be  stated  by  the 
pupils  under  the  teacher's  guidance. 

The  definition  of  addition,  for  illustration,  ma}'  be 
taught  by  writing  several  examples  on  the  blackboard 
(using  small  concrete  numbers),  and  by  questions  lead- 
ing the  pupils  to  see  that  the  sum  contains  as  many 
units  or  ones  as  all  the  numbers  added  taken  together. 
The  general  principle  may  thus  be  stated : 


MISCELLANEOUS   LESSONS.  187 

The  sum  of  two  or  more  numbers  contains  as  many  units 
as  all  the  given  numbers. 

The  fact  that  addition  is  a  process  may  next  be  taught, 
and  the  pupils  then  led  to  the  following  definition  : 

Addition  is  the  process  of  finding  the  sum  of  two  or  more 
numbers. 

NOTE.— More  advanced  pupils  may  be  shown  that  the  facts  in 
the  above  principle  and  definition  may  be  united  in  one  sen- 
tence, as  follows: 

Addition  is  t)ie  process  of  finding  a  number  (hat  contains  as  many 
units  as  two  or  more  given  numbers. 

In  elementary  classes,  it  is  believed  to  be  better  to 
define  the  term  sum  (as  above),  and  define  addition  by 
the  use  of  this  known  term. 


SUMMARY. 

The  above  suggestions  combined  give  the  following 
order  for  teaching  each  general  process  in  elementary 
arithmetic  : 

1.  The  oral  drill  on  the  inductive  oral  exercises. 

2.  The  induction  of  the  written  process  from  the  oral 
solutions,  under  the  teacher's  guidance. 

3.  The   solution    of    the    oral    problems    on    slate    or 
paper  by  the  written  process. 

4.  The  solution  of  the  written  problems  on  slate  or 
paper. 

5.  The  induction  of  the  rule  from  the  written  process, 
and  the  memorizing  of  the  approved  rule. 

6.  The  induction  and  memorizing  of  definitions  and 
principles. 


WH  ITE'S 

NEW   ARITHMETICS. 


TWO-BOOK  SERIES. 


Uniting  Oral  and  Written  Processes,  and  Embodying  the  Inductive  Method. 


WHITE'S  NEW  ELEMENTARY  ARITHMETIC. 
WHITE'S  NEW  COMPLETE  ARITHMETIC. 


White's  new  series  of  Arithmetics  is  designed  to  meet 
the  demand  for  a  two-book  series,  presenting  a  progressive 
and  complete  arithmetical  course  for  school  use. 

A  first  book  in  Arithmetic  is  obviously  defective, 
which  ends  too  low  or  begins  too  high,  and  hence  does  not 
present  an  adequate  and  satisfactory  elementary  course. 
The  use  of  too  brief  a  first  book  brings  pupils  to  the 
second  or  advanced  book  at  too  early  an  age,  since  nearly 
every  page  of  such  a  treatise  presents  matter  too  diffi- 
cult for  young  children.  On  the  other  hand,  the  omis- 
sion of  the  primary  lessons  in  number,  given  during 
the  first  four  years  of  school  instruction,  puts  on  teach- 
ers a  large  amount  of  unnecessary  labor  in  preparing 
blackboard  and  slate  work,  while  the  amount  of  copy- 
ing required  of  the  pupils  is  a  severe  tax  on  the  eyes 
and  nerves  of  little  children. 

(189) 


190  WHITE'S  NEW  ARITHMETICS. 

WHITE'S  NEW  ELEMENTARY  ARITHMETIC  avoids  both 
of  these  errors,  and  is  believed  to  be  the  only  first  book 
of  Arithmetic  yet  published  that  covers  the  entire 
ground  of  elementary  instruction,  and  affords  an  ade- 
quate preparation  for  the  more  advanced  treatise.  It 
presents  a  series  of  progressive  and  carefully  graded 
exercises  for  the  mastery  of  all  elementary  processes 
with  integral  and  fractional  numbers,  and  these  exer- 
cises are  full  enough  to  bear  the  test  of  actual  use  in  ele- 
mentary schools. 

White's  New  Elementary  Arithmetic  also  includes  those 
practical  applications  of  the  elementary  processes  which  are 
most  frequently  used  in  business  and  common  life.  All 
these  applications,  including  United  States  Money, 
Mensuration,  Percentage,  Simple  Interest,  and  Discount, 
are  presented  in  a  simple  manner,  and  the  problems 
are  within  the  comprehension  of  young  pupils.  It  is 
thus  seen  that  while  the  New  Elementary  prepares 
pupils  for  the  successful  study  of  the  second  book  (the 
New  Complete),  it  is  in  itself  a  practical  One-book 
Course  in  Arithmetic  for  the  large  number  of  pupils,  in 
both  city  and  country,  who  do  not  attend  school  long  enough 
to  reach  Percentage  and  other  practical  applications  of  num- 
ber in  the  higher  and  more  extensive  treatise. 

WHITE'S  NEW  COMPLETE  ARITHMETIC  retains  all  the 
excellent  features  which  have  made  White's  Complete 
Arithmetic  so  strong  and  popular  a  text-book,  with  an 
increase  of  business  problems,  and  a  fuller  and  better 
treatment  of  the  Metric  System,  Mensuration,  Stock  In- 
vestments, and  other  practical  subjects. 

Special  attention  is  called  to  the  following  important 
features  which  characterize  White's  New  Two-book  Series: 

1.  Thorough  and  varied  drills  in  all  fundamental  pro- 
cesses. The  series  not  only  presents  a  sufficient  number 
of  problems  under  each  process  to  enable  pupils  to 
master  it,  but  it  gives  a  large  number  of  miscellaneous 


WHITE'S  NEW  ARITHMETICS.  191 

revieto  problems,  in  the  solution  of  which  pupils  must 
determine  the  process  or  processes  to  be  used.  All  ex- 
perienced teachers  of  Arithmetic  know  that  pupils  may 
be  able  to  solve  readily  problems  which  are  so  classified 
that  each  series  is  solved  by  one  process,  and  yet  fail 
badly  when  required  to  solve  miscellaneous  problems, 
involving  different  processes,  to  be  determined  by  a  care- 
ful study  of  each  problem.  White's  New  Two-book 
Series  contains  an  unusually  large  number  of  miscel- 
laneous and  review  problems. 

2.  The  presentation   of   the  current  values,  forms,    and 
usages  of  business,  and  the  large  number  and  variety  of 
business  problems.     The  series  abounds  in  practical  appli- 
cations  of   numerical   processes  to  mercantile  business, 
the    mechanic   arts,    domestic    life,    etc.      A    glance    at 
United  States  Money,  Mensuration,  Percentage,  etc.,  in 
both  books,  will  suffice  to  show  that  this  is  a  promi- 
nent feature  of  the  series. 

3.  The  number  and  variety  of  oral  problems,  and   espe- 
cially of  review  problems  for  oral  analysis.     White's  series 
of  Arithmetics  unite  oral  and   written   exercises   in  a 
practiced  and  philosophical  manner;    and   the   practical 
value  of  this  innovation  is  attested  by  its  general  adop- 
tion in  modern  Arithmetics.     White's  Arithmetics  not 
only  contain  admirable  oral  inductive  exercises,  but,  in 
addition,  they  present  series  of  problems  for  oral  analysis, 
thus  preserving  the  advantages  resulting  from  the  oral 
analytic  drills  which  characterize  what  is  called  "  men- 
tal arithmetic."     The  introduction  of  these  oral  analytic 
drills  over  forty  years  ago  revolutionized  the  methods 
of    teaching   Arithmetic   then    in    use.      White's    series 
contains  as  many   and  as  great  a  variety  of  problems 
for  oral  analysis  as  the  ordinary  separate  mental  arith- 
metics, but  these  oral  inductive  exercises  and  analytic 
drills  accompany  the  written  processes,  and  hence  are  not 
reached  by  pupils  before  they  are  prepared  to  profit  by 


192  WHITE'S  NEW  ARITHMETICS. 

them,  which  was  often  not  the  case  when  pupils  were 
required  to  complete  the  so-called  Mental  Arithmetic 
before  beginning  the  study  of  Written  Arithmetic. 

4.  Another  important  feature  is  a  true  and  practical 
embodiment    of    the    inductive    method    of    teaching.      The 
written   processes   are   not   only   reached    by   inductive 
oral  exercises,  but  the  definitions,  principles,  and  rules 
follow  processes  and  problems — a  characteristic  feature 
that  has  widely  commended  the  series  to  the  most  pro- 
gressive  and   thorough   teachers   of  Arithmetic  in  the 
country. 

5.  The  Definitions,  Rules,  and  Statements  of  Principles 
are  models  of  accuracy,  clearness,  and  conciseness. 


VAN  ANTWERP,  BRAGG  &  CO.,    PUBLISHERS, 

CINCINNATI    AND    NEW    YORK. 


VB  17256 


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